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# On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype

Alessandro Carderi, Damien Gaboriau, François Le Maître, Yves Stalder

► To cite this version:

Alessandro Carderi, Damien Gaboriau, François Le Maître, Yves Stalder. On the space of subgroups of Baumslag-Solitar groups I: perfect kernel and phenotype. 2022. hal-03829832v2

HAL Id: hal-03829832

https://hal.science/hal-03829832v2

Preprint submitted on 26 Oct 2022

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<table>
  <tbody>
    <tr>
      <td>1.6</td>
      <td>Some ideas on the techniques of proofs</td>
      <td>10</td>
    </tr>
    <tr>
      <td>1.7</td>
      <td>Subsequent work</td>
      <td>12</td>
    </tr>
    <tr>
      <td><b>2</b></td>
      <td><b>Preliminaries and notations</b></td>
      <td><b>12</b></td>
    </tr>
    <tr>
      <td>2.1</td>
      <td>Graphs and Schreier graphs</td>
      <td>13</td>
    </tr>
    <tr>
      <td>2.2</td>
      <td>Space of subgroups</td>
      <td>14</td>
    </tr>
    <tr>
      <td>2.3</td>
      <td>Bass-Serre theory</td>
      <td>17</td>
    </tr>
    <tr>
      <td><b>3</b></td>
      <td><b>Bass-Serre graphs</b></td>
      <td><b>20</b></td>
    </tr>
    <tr>
      <td>3.1</td>
      <td>Pre-actions</td>
      <td>20</td>
    </tr>
    <tr>
      <td>3.2</td>
      <td>Bass-Serre graphs</td>
      <td>21</td>
    </tr>
    <tr>
      <td>3.3</td>
      <td>(m, n)-graphs</td>
      <td>24</td>
    </tr>
    <tr>
      <td>3.4</td>
      <td>Realizing (m, n)-graphs as Bass-Serre graphs</td>
      <td>26</td>
    </tr>
    <tr>
      <td>3.5</td>
      <td>Additional properties of (m, n)-graphs</td>
      <td>28</td>
    </tr>
    <tr>
      <td>3.6</td>
      <td>Bass-Serre theory for BS(m, n)</td>
      <td>29</td>
    </tr>
    <tr>
      <td><b>4</b></td>
      <td><b>Phenotype</b></td>
      <td><b>31</b></td>
    </tr>
    <tr>
      <td>4.1</td>
      <td>Phenotypes of natural numbers</td>
      <td>31</td>
    </tr>
    <tr>
      <td>4.2</td>
      <td>Phenotypes of (m, n)-graphs</td>
      <td>33</td>
    </tr>
    <tr>
      <td>4.3</td>
      <td>Phenotypes of BS(m, n)-actions</td>
      <td>34</td>
    </tr>
    <tr>
      <td>4.4</td>
      <td>Merging pre-actions</td>
      <td>35</td>
    </tr>
    <tr>
      <td><b>5</b></td>
      <td><b>Perfect kernel and dense orbits</b></td>
      <td><b>40</b></td>
    </tr>
    <tr>
      <td>5.1</td>
      <td>Perfect kernels of Baumslag-Solitar groups</td>
      <td>40</td>
    </tr>
    <tr>
      <td>5.2</td>
      <td>Phenotypical decomposition of the perfect kernel</td>
      <td>43</td>
    </tr>
    <tr>
      <td>5.3</td>
      <td>Closed invariant subsets with a fixed finite phenotype</td>
      <td>46</td>
    </tr>
    <tr>
      <td><b>6</b></td>
      <td><b>Limits of finite phenotype subgroups</b></td>
      <td><b>52</b></td>
    </tr>
    <tr>
      <td>6.1</td>
      <td>Limits of subgroups with fixed finite phenotype</td>
      <td>52</td>
    </tr>
    <tr>
      <td>6.2</td>
      <td>Limits of subgroups with varying finite phenotype</td>
      <td>55</td>
    </tr>
  </tbody>
</table>

# 1 Introduction and presentation of the results

The Baumslag-Solitar group of non-zero integer parameters $m$ and $n$ is defined by the presentation

$$ \mathrm{BS}(m, n) := \langle b, t | tb^m t^{-1} = b^n \rangle. \qquad (1.1) $$
<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>2 Preliminaries and notations</td><td>11</td></tr><tr><td>2.1 Graphs and Schreier graphs</td><td>11</td></tr><tr><td>2.2 Space of subgroups</td><td>13</td></tr><tr><td>2.3 Bass-Serre theory</td><td>15</td></tr><tr><td>3 Bass-Serre graphs</td><td>19</td></tr><tr><td>3.1 Pre-actions</td><td>19</td></tr><tr><td>3.2 Bass-Serre graphs</td><td>20</td></tr><tr><td>3.3 (m, n)-graphs</td><td>22</td></tr><tr><td>3.4 Realizing (m, n)-graphs as Bass-Serre graphs</td><td>24</td></tr><tr><td>3.5 Additional properties of (m, n)-graphs</td><td>26</td></tr><tr><td>3.6 Bass-Serre graphs and Bass-Serre theory</td><td>27</td></tr><tr><td>4 Phenotype</td><td>29</td></tr><tr><td>4.1 Phenotypes of natural numbers</td><td>30</td></tr><tr><td>4.2 Phenotypes of (m, n)-graphs</td><td>31</td></tr><tr><td>4.3 Phenotypes of BS(m, n)-actions</td><td>32</td></tr><tr><td>4.4 Merging pre-actions</td><td>33</td></tr><tr><td>5 Perfect kernel and dense orbits</td><td>38</td></tr><tr><td>5.1 Perfect kernels of Baumslag-Solitar groups</td><td>38</td></tr><tr><td>5.2 Phenotypical decomposition of the perfect kernel</td><td>41</td></tr><tr><td>5.3 Closed invariant subsets with a fixed finite phenotype</td><td>44</td></tr><tr><td>6 Limits of finite phenotype subgroups</td><td>49</td></tr><tr><td>6.1 Limits of subgroups with fixed finite phenotype</td><td>50</td></tr><tr><td>6.2 Limits of subgroups with finite phenotype</td><td>53</td></tr></tbody></table>

# 1 Introduction and presentation of the results

The Baumslag-Solitar group of non-zero integer parameters $m$ and $n$ is defined by the presentation

$$ \mathrm{BS}(m, n) := \langle b, t | tb^m t^{-1} = b^n \rangle. \qquad (1.1) $$

These one-relator two-generators groups were defined by Baumslag and Solitar [BS62] to provide examples of groups with surprising properties, depending on the arithmetic properties of the parameters.
These one-relator two-generators groups were defined by Baumslag and Solitar [BS62] to provide examples of groups with surprising properties, depending on the arithmetic properties of the parameters.

It results from the work of Baumslag and Solitar and of Meskin [Mes72] that the group $BS(m, n)$ is

* residually finite if and only if $|m| = 1$ or $|n| = 1$ or $|m| = |n|$;

* Hopfian if and only if it is residually finite or $m$ and $n$ have the same set of prime divisors.

The group $BS(m, n)$ is amenable if and only if $|m| = 1$ or $|n| = 1$, and in this case, it is metabelian. All Baumslag-Solitar groups however share weak forms of amenability: they are inner-amenable [Sta06] and a-T-menable [GJ03].

Over the years and despite the simplicity of their presentation, these groups have served as a standard source of examples and counter-examples, sometimes to published results (!). They have been considered from countless different perspectives in group theory and beyond.

Various aspects concerning the subgroups of the $BS(m, n)$ have been considered such as the growth functions of their number of subgroups of finite index with various properties, or such as a description of the kind of fundamental group of graphs of groups that can be embedded as subgroups in some $BS(m, n)$; see for instance [Gel05, Dud09, Lev15].

In this article, we consider global aspects of the space $Sub(BS(m, n))$ of subgroups of the $BS(m, n)$ and of the topological dynamics generated by the natural action by conjugation.

## 1.1 The perfect kernel

Let $\Gamma$ be a countable group. We denote by $Sub(\Gamma)$ the space of subgroups of $\Gamma$. If one identifies each subgroup with its indicator function, one can view the space $Sub(\Gamma)$ as a closed subset of $\{0, 1\}^{\Gamma}$. Thus $Sub(\Gamma)$ is a compact, metrizable space by giving it the restriction of the product topology. See Section 2.2 for the generalities about $Sub(\Gamma)$.

By the Cantor-Bendixson theorem, $Sub(\Gamma)$ admits a unique decomposition as a disjoint union of a perfect set, called the **perfect kernel** $K(\Gamma)$ of $\Gamma$, and of a countable open subset. It is a challenging problem to determine the perfect kernel of a given countable group.

When $\Gamma$ is finitely generated, the finite index subgroups are isolated in $Sub(\Gamma)$. It is thus relevant to introduce the subspace $Sub_{[\infty]}(\Gamma)$ consisting of
It results from the work of Baumslag and Solitar and of Meskin [Mes72]
that the group $\text{BS}(m, n)$ is

• residually finite if and only if $|m| = 1$ or $|n| = 1$ or $|m| = |n|$;

• Hopfian if and only if it is residually finite or *m* and *n* have the same set of prime divisors.

The group $\text{BS}(m, n)$ is amenable if and only if $|m| = 1$ or $|n| = 1$, and in this case, it is metabelian. All Baumslag-Solitar groups however share weak forms of amenability: they are inner-amenable [Sta06b] and a-T-menable [GJ03].

Over the years and despite the simplicity of their presentation, these
groups have served as a standard source of examples and counter-examples,
sometimes to published results (!). They have been considered from countless
different perspectives in group theory and beyond.

Various aspects concerning the subgroups of the $\text{BS}(m, n)$ have been considered such as the growth functions of their number of subgroups of finite index with various properties, or such as a description of the kind of fundamental group of graphs of groups that can be embedded as subgroups in some $\text{BS}(m, n)$; see for instance [Gel05, Dud09, Lev15].

In this article, we consider global aspects of the space $\text{Sub}(\text{BS}(m, n))$ of subgroups of the $\text{BS}(m, n)$ and of the topological dynamics generated by the natural action by conjugation.

1.1 The perfect kernel

Let $\Gamma$ be a countable group. We denote by $\text{Sub}(\Gamma)$ the space of subgroups of $\Gamma$. If one identifies each subgroup with its indicator function, one can view the space $\text{Sub}(\Gamma)$ as a closed subset of $\{0, 1\}^{\Gamma}$. Thus $\text{Sub}(\Gamma)$ is a compact, metrizable space by giving it the restriction of the product topology. See Section 2.2 for the generalities about $\text{Sub}(\Gamma)$.

By the Cantor–Bendixson theorem, $\text{Sub}(\Gamma)$ admits a unique decomposition as a disjoint union of a perfect set, called the **perfect kernel** $\mathcal{K}(\Gamma)$ of $\Gamma$, and of a countable open subset. It is a challenging problem to determine the perfect kernel of a given countable groups.

When $\Gamma$ is finitely generated, the finite index subgroups are isolated in $\text{Sub}(\Gamma)$. It is thus relevant to introduce the subspace $\text{Sub}_{[\infty]}(\Gamma)$ consisting of all infinite index subgroups of $\Gamma$. It is a closed subspace of $\text{Sub}(\Gamma)$ exactly when $\Gamma$ is finitely generated (see Remark 2.3).
all infinite index subgroups of $\Gamma$. It is a closed subspace of $\text{Sub}(\Gamma)$ exactly when $\Gamma$ is finitely generated (see Remark 2.3).

Our first main result completely describes the perfect kernel of the various Baumslag-Solitar groups. When $|m| = |n|$, the subgroup generated by $b^m$ is normal; let us denote by $\pi$ the corresponding quotient homomorphism

$$ \mathrm{BS}(m, n) \xrightarrow{\pi} \mathrm{BS}(m, n)/\langle b^m \rangle. $$

We also denote by $\pi$ the map it induces between the spaces of subgroups of $\mathrm{BS}(m, n)$ and $\mathrm{BS}(m, n)/\langle b^m \rangle$.

**Theorem A** (Perfect kernel of $\mathrm{BS}(m, n)$, Theorem 5.3). Let $m, n \in \mathbb{Z} \setminus \{0\}$,

1. if $|m| = 1$ or $|n| = 1$, then $\mathcal{K}(\mathrm{BS}(m, n))$ is empty;

2. if $|m|, |n| > 1$, then

(a) if $|m| \neq |n|$, then $\mathcal{K}(\mathrm{BS}(m, n)) = \text{Sub}_{[\infty]}(\mathrm{BS}(m, n))$;

(b) if $|m| = |n|$, then $\mathcal{K}(\mathrm{BS}(m, n)) = \pi^{-1}(\text{Sub}_{[\infty]}(\mathrm{BS}(m, n)/\langle b^m \rangle))$.

The fact that $\text{Sub}(\mathrm{BS}(m, n))$ is countable when $|m| = 1$ or $|n| = 1$ (Item 1), i.e. for the Baumslag-Solitar groups that are metabelian, was already observed by Becker, Lubotzky, and Thom [BLT19, Corollary 8.4]. Fortuitously or not, it turns out that $\mathcal{K}(\mathrm{BS}(m, n)) = \text{Sub}_{[\infty]}(\mathrm{BS}(m, n))$ exactly when $\mathrm{BS}(m, n)$ is not residually finite.

There is a general correspondence between the transitive pointed $\Gamma$-actions and the subgroups of $\Gamma$. It sends an action $\alpha$ to the stabilizer of its base point. This $\Gamma$-equivariant map is a bijection when one considers the actions up to pointed isomorphisms (see Section 2.2). Item 2 of Theorem A has a unified reformulation in this setting:

2'. if $|m|, |n| > 1$, then $\mathcal{K}(\mathrm{BS}(m, n))$ is the space of subgroups $\Lambda$ such that the right $\mathrm{BS}(m, n)$-action on $\Lambda \setminus \mathrm{BS}(m, n)$ has infinitely many $\langle b \rangle$-orbits.

Note that this exactly means that the quotient of the $\Lambda$-action on the standard Bass-Serre tree (see Section 2.3) of $\mathrm{BS}(m, n)$ is infinite.

Let us now give some more context for Theorem A. By Brouwer's characterization of Cantor spaces, the space $\mathcal{K}(\Gamma)$ is either empty or a Cantor space. It is empty exactly when $\text{Sub}(\Gamma)$ is countable. This happens for example for groups all whose subgroups are finitely generated, also known as Noetherian groups. For instance all finitely generated nilpotent groups and more generally all polycyclic groups have a countable space of subgroups.

On the opposite side, for the free group with a countably infinite number of generators, no subgroup is isolated, thus $\mathcal{K}(\mathbf{F}_\infty) = \text{Sub}(\mathbf{F}_\infty)$ (see [CGLM23, Proposition 2.1]).
Our first main result completely describes the perfect kernel of the various Baumslag-Solitar groups. When $|m| = |n|$, the subgroup generated by $b^m$ is normal; let us denote by $\pi$ the corresponding quotient homomorphism

$$BS(m, n) \xrightarrow{\pi} BS(m, n)/\langle b^m \rangle.$$

**Theorem A** (Perfect kernel of $BS(m, n)$, Theorem 5.3). Let $m, n \in \mathbb{Z}\setminus\{0\}$,

1. if $|m| = 1$ or $|n| = 1$, then $\mathcal{K}(BS(m, n))$ is empty;

2. if $|m|, |n| > 1$, then

(a) if $|m| \neq |n|$, then $\mathcal{K}(BS(m, n)) = \text{Sub}_{[\infty]}(BS(m, n))$;

(b) if $|m| = |n|$, then $\mathcal{K}(BS(m, n)) = \pi^{-1}(\text{Sub}_{[\infty]}(BS(m, n)/\langle b^m \rangle))$.

The fact that $\text{Sub}(BS(m, n))$ is countable when $|m| = 1$ or $|n| = 1$ (Item 1), i.e. for the Baumslag-Solitar groups that are metabelian, was already observed by Becker, Lubotzky, and Thom [BLT19, Corollary 8.4]. Fortuitously or not, it turns out that $\mathcal{K}(BS(m, n)) = \text{Sub}_{[\infty]}(BS(m, n))$ exactly when $BS(m, n)$ is not residually finite.

There is a general correspondence between the transitive pointed $\Gamma$-actions and the subgroups of $\Gamma$. It sends an action $\alpha$ to the stabilizer of its base point. This $\Gamma$-equivariant map is a bijection when one considers the actions up to pointed isomorphisms (see Section 2.2). Item 2 of Theorem A has a unified reformulation in this setting:

2'. if $|m|, |n| > 1$, then $\mathcal{K}(BS(m, n))$ is the space of subgroups $\Lambda$ such that the right $BS(m, n)$-action on $\Lambda\setminus BS(m, n)$ has infinitely many $\langle b \rangle$-orbits. Note that this exactly means that the quotient of the $\Lambda$-action on the standard Bass-Serre tree (see Section 2.3) of $BS(m, n)$ is infinite.

Let us now give some more context for Theorem A. By Brouwer's characterization of Cantor spaces, the space $\mathcal{K}(\Gamma)$ is either empty or a Cantor space. It is empty exactly when $\text{Sub}(\Gamma)$ is countable. This happens for example for groups all whose subgroups are finitely generated, also known as Noetherian groups. For instance all finitely generated nilpotent groups and more generally all polycyclic groups have a countable space of subgroups.

On the opposite side, for the free group with a countably infinite number of generators, no subgroup is isolated, thus $\mathcal{K}(F_\infty) = \text{Sub}(F_\infty)$ (see [CGLM22, Proposition 2.1]).

There are some classical groups for which we know that $\mathcal{K}(\Gamma) = \text{Sub}_{[\infty]}(\Gamma)$. This is the case for the free groups $F_n$ (for $1 < n < \infty$), see for instance [CGLM22, Proposition 2.1]. This is also the case for the groups with infinitely many ends, for the fundamental groups of the closed surfaces of genus $\ge 2$,
There are some classical groups for which we know that $\mathcal{K}(\Gamma) = \text{Sub}_{[\infty]}(\Gamma)$. This is the case for the free groups $F_n$ (for $1 < n < \infty$), see for instance [CGLM23, Proposition 2.1]. This is also the case for the groups with infinitely many ends, for the fundamental groups of the closed surfaces of genus $\ge 2$, and for the finitely generated LERF groups with non-zero first $\ell^2$-Betti number (see [AG23]). Recall that a group $\Gamma$ is LERF when its set of finite index subgroups is dense in $\text{Sub}(\Gamma)$ (see for instance [GKM16, Theorem 3.1]).

Bowen, Grigorchuk and Kravchenko established that the perfect kernel of the lamplighter group $(\mathbb{Z}/p\mathbb{Z})^n \wr \mathbb{Z} = (\bigoplus_{\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z})^n) \times \mathbb{Z}$ (for a prime number $p$) is exactly the space $\text{Sub}(\bigoplus_{\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z})^n)$ of subgroups of the normal subgroup [BGK15, Theorem 1.1]. Skipper and Wesolek uncovered the perfect kernel for a class of branch groups containing the Grigorchuk group and the Gupta-Sidki 3 group [SW20].

The perfect kernel can be obtained by successively, and transfinitely, removing the isolated points, thus obtaining for every ordinal $\alpha$ the $\alpha$-th Cantor-Bendixson derivative $\text{Sub}(\Gamma)^{(\alpha)} := \text{Sub}(\Gamma)^{(\beta)} \setminus \{\text{isolated points of } \text{Sub}(\Gamma)^{(\beta)}\}$ if $\alpha = \beta + 1$, and $\text{Sub}(\Gamma)^{(\alpha)} := \bigcap_{\beta<\alpha} \text{Sub}(\Gamma)^{(\beta)}$ if $\alpha$ is a limit ordinal. The **Cantor-Bendixson rank** $\text{rk}_{CB}(\Gamma)$ of $\Gamma$ is the first ordinal $\zeta$ for which the derived space $\text{Sub}(\Gamma)^{(\zeta)}$ has no more isolated points, and is thus equal to the perfect kernel (see for instance [Kec95, Section 6.C] for details). When $|m|, |n| > 1$ and $|m| \ne |n|$, then Theorem A implies that $\text{rk}_{CB}(\text{BS}(m, n)) = 1$. The determination of the Cantor-Bendixson ranks $\text{rk}_{CB}(\text{BS}(m, n))$ for the other cases is postponed to the sequel [CGLMS].

## 1.2 Dynamical partition of the perfect kernel

The compact space of subgroups $\text{Sub}(\Gamma)$ is equipped with the continuous action of $\Gamma$ by conjugation: $\Lambda \cdot \gamma := \gamma^{-1}\Lambda\gamma$. The perfect kernel is $\Gamma$-invariant. This action is of course not minimal in general, even when restricted to the perfect kernel: the latter may contain normal subgroups and these subgroups are fixed points! However, the first three named authors observed a particularly nice feature in the case of the free group $F_n$ (for $1 < n < \infty$): the action $\mathcal{K}(F_n) \hookrightarrow F_n$ is topologically transitive (which means that the space admits a dense $G_\delta$ subset of points whose individual orbits are dense). These $F_n$-actions are called totipotent, see [CGLM23].

To our surprise, we uncovered a dramatically different and very rich situation for the Baumslag-Solitar groups.
and for the finitely generated LERF groups with non-zero first $l^2$-Betti number (see [AG22]). Recall that a group $\Gamma$ is LERF when its set of finite index subgroups is dense in $\text{Sub}(\Gamma)$ (see for instance [GKM16, Theorem 3.1]).

Bowen, Grigorchuk and Kravchenko established that the perfect kernel of the lamplighter group $(\mathbb{Z}/p\mathbb{Z})^n \ltimes \mathbb{Z} = (\bigoplus_{\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z})^n) \times \mathbb{Z}$ (for a prime number $p$) is exactly the space $\text{Sub}(\bigoplus_{\mathbb{Z}}(\mathbb{Z}/p\mathbb{Z})^n)$ of subgroups of the normal subgroup [BGK15, Theorem 1.1]. Skipper and Wesolek uncovered the perfect kernel for a class of branch groups containing the Grigorchuk group and the Gupta-Sidki 3 group [SW20].

The perfect kernel can be obtained by successively, and transfinitely, removing the isolated points. The **Cantor-Bendixson rank** $rk_{CB}(\Gamma)$ of $\Gamma$ is the first ordinal $\zeta$ for which the derived space $\text{Sub}(\Gamma)^{(\zeta)}$ has no more isolated points. When $|m|, |n| > 1$ and $|m| \neq |n|$, then Theorem A implies that $rk_{CB}(\text{BS}(m, n)) = 1$. The determination of the Cantor-Bendixson ranks $rk_{CB}(\text{BS}(m, n))$ for the other cases is postponed to the sequel [CGLMS23].

## 1.2 Dynamical partition of the perfect kernel

The compact space of subgroups $\text{Sub}(\Gamma)$ is equipped with the continuous action of $\Gamma$ by conjugation: $\gamma \cdot \Lambda := \gamma\Lambda\gamma^{-1}$. The perfect kernel is $\Gamma$-invariant. This action is of course not minimal in general, even when restricted to the perfect kernel: the latter may contain normal subgroups and these subgroups are fixed points! However, the first three named authors observed a particularly nice feature in the case of the free group $F_n$ (for $1 < n < \infty$): the action $F_n \to \mathcal{K}(F_n)$ is topologically transitive (which means that the space admits a dense $G_\delta$ subset of points whose individual orbits are dense). These $F_n$-actions are called totipotent, see [CGLM22].

To our surprise, we uncovered a dramatically different and very rich situation for the Baumslag-Solitar groups.

**Theorem B.** Whenever $|m|, |n| \neq 1$, the perfect kernel $\mathcal{K}(\text{BS}(m, n))$ admits a countably infinite partition into $\text{BS}(m, n)$-invariant and topologically transitive subspaces. One of them is closed; all the other ones are open (for the induced topology).

Theorem B follows from Proposition 5.8 and Theorem 5.14. From now on in this introduction, we stick to the case $|m| \neq 1$ and $|n| \neq 1$. In order to describe the partition in Theorem B, we introduce a new invariant: the **phenotype**.
**Theorem B.** Whenever $|m|, |n| \neq 1$, the perfect kernel $\mathcal{K}(\mathrm{BS}(m, n))$ admits a countably infinite partition into $\mathrm{BS}(m, n)$-invariant and topologically transitive subspaces. For the induced topology on $\mathcal{K}(\mathrm{BS}(m, n))$, one of the subspaces is closed; all the other ones are open.

Theorem B follows from Proposition 5.8 and Theorem 5.14. In a further work [GLMS24], we show that topological transitivity can be upgraded to high topological transitivity.

From now on in this introduction, we stick to the case $|m| \neq 1$ and $|n| \neq 1$. In order to describe the partition in Theorem B, we introduce a new invariant: the **phenotype**.

The relation $tb^mb^{-1} = b^n$ imposes some arithmetic conditions between the cardinalities of the $b$-orbit of a point $x$ and the $b$-orbit of $xt$. For instance, the $b$-orbit of $x$ is infinite if and only if the $b$-orbit of $xt$ is infinite.

In Definition 4.1, we introduce a function $\mathrm{Ph}_{m,n}: \mathbb{Z}_{\geq 1} \cup \{\infty\} \to \mathbb{Z}_{\geq 1} \cup \{\infty\}$ called the $(m, n)$-phenotype, with the following property, which directly follows from Proposition 4.6, Theorem 4.13 and Proposition 3.22:

**Theorem C.** Whenever $|m|, |n| \neq 1$, there is a transitive $\mathrm{BS}(m, n)$-action with two $b$-orbits of cardinal $k$ and $\ell$ respectively if and only if $\mathrm{Ph}_{m,n}(k) = \mathrm{Ph}_{m,n}(\ell)$.

If for instance $m$ and $n$ are coprime, the phenotype $\mathrm{Ph}_{m,n}(k)$ of any natural number $k \in \mathbb{Z}_{\geq 1}$ is obtained as $k$ expunged of all its prime divisors that appear in either $m$ or $n$. The general form is more complicated, see Definition 4.1 and Example 4.3, but it follows readily from Definition 4.1 that $\mathrm{Ph}_{m,n}(q) = q$ for every $q \geq 1$ that is coprime with $m$ and $n$. Hence, the set of possible $(m, n)$-phenotypes

$$Q_{m,n} := \{\mathrm{Ph}_{m,n}(k) : k \in \mathbb{Z}_{\geq 1}\} \cup \{\infty\}.$$

is always infinite.

Theorem C allows us to define the **phenotype** of a transitive $\mathrm{BS}(m, n)$-action as the common $(m, n)$-phenotype of the cardinalities of its $b$-orbits. Then, we define, the **phenotype $\mathrm{Ph}(\Lambda)$** of a subgroup $\Lambda \in \mathrm{Sub}(\mathrm{BS}(m, n))$ as the phenotype of the (right) $\mathrm{BS}(m, n)$-action on the homogeneous space $\Lambda\backslash\mathrm{BS}(m, n)$.

Notice that the $\mathrm{BS}(m, n)$-actions on $\Lambda\backslash\mathrm{BS}(m, n)$ and $(g^{-1}\Lambda g)\backslash\mathrm{BS}(m, n)$ are isomorphic (both are transitive with some point stabilizer equal to $\Lambda$), so
The relation $tb^mb^{-1} = b^n$ imposes some arithmetic conditions between the cardinalities of the b-orbit of a point $x$ and the b-orbit of $xt$. For instance, the b-orbit of $x$ is infinite if and only if the b-orbit of $xt$ is infinite.

In Definition 4.1, we introduce a function $Ph_{m,n}: \mathbb{Z}_{\ge 1} \cup \{\infty\} \to \mathbb{Z}_{\ge 1} \cup \{\infty\}$ called the $(m, n)$-phenotype, with the following property, which directly follows from Proposition 4.6, Theorem 4.13 and Proposition 3.22:

**Theorem C.** Whenever $|m|, |n| \ne 1$, there is a transitive $\text{BS}(m, n)$-action with two b-orbits of cardinal $k$ and $l$ respectively if and only if $Ph_{m,n}(k) = Ph_{m,n}(l)$.

If for instance $m$ and $n$ are coprime, the phenotype $Ph_{m,n}(k)$ of any natural number $k \in \mathbb{Z}_{\ge 1}$ is obtained as $k$ expunged of all its prime divisors that appear in either $m$ or $n$. The general form is more complicated, see Definition 4.1 and Example 4.3, but it follows readily from Definition 4.1 that $Ph_{m,n}(q) = q$ for every $q \ge 1$ that is coprime with $m$ and $n$. Hence, the set of possible $(m, n)$-phenotypes

$$ \mathcal{Q}_{m,n} := \{\text{Ph}_{m,n}(k) : k \in \mathbb{Z}_{\ge 1}\} \cup \{\infty\}. $$

is always infinite.

Theorem C allows us to define the **phenotype** of a transitive $\text{BS}(m, n)$-action as the common $(m, n)$-phenotype of the cardinalities of its b-orbits. Then, we define, the **phenotype** $\text{Ph}(\Lambda)$ of a subgroup $\Lambda \in \text{Sub}(\text{BS}(m, n))$ as the phenotype of its action on the homogeneous space $\Lambda \backslash \text{BS}(m, n)$.

Notice that the $\text{BS}(m, n)$-actions on $\Lambda \backslash \text{BS}(m, n)$ and $(g\Lambda g^{-1}) \backslash \text{BS}(m, n)$ are isomorphic (both are transitive with some point stabilizer equal to $\Lambda$), so that they have the same phenotype. Hence, the partition

$$ \text{Sub}(\text{BS}(m, n)) = \bigsqcup_{q \in \mathcal{Q}_{m,n}} \text{Ph}^{-1}(q) \qquad (1.2) $$

is invariant under the $\text{BS}(m, n)$-action (recall this is the action by conjugation). Let us mention from Proposition 5.8 that

* for all finite $q \in \mathcal{Q}_{m,n}$, the pieces $\text{Ph}^{-1}(q)$ are open;

* the piece $\text{Ph}^{-1}(\infty)$ is closed but not open.

It now follows from Theorem 5.14 that: the restriction of the partition (1.2) to the perfect kernel

$$ \mathcal{K}(\text{BS}(m, n)) = \bigsqcup_{q \in \mathcal{Q}_{m,n}} \mathcal{K}_q(\text{BS}(m, n)), \qquad (1.3) $$
they have the same phenotype. Hence, the partition

$$ \mathrm{Sub}(\mathrm{BS}(m, n)) = \bigsqcup_{q \in \mathbb{Q}_{m,n}} \mathbf{Ph}^{-1}(q) \qquad (1.2) $$

is invariant under the $\mathrm{BS}(m, n)$-action (recall this is the action by conjugation). Let us mention from Proposition 5.8 that

* for each finite $q \in \mathbb{Q}_{m,n}$, the piece $\mathbf{Ph}^{-1}(q)$ is open; it is also closed if and only if $|m| = |n|$;

* the piece $\mathbf{Ph}^{-1}(\infty)$ is closed but not open.

In particular, the function $\mathbf{Ph}$: $\mathrm{Sub}(\mathrm{BS}(m, n)) \to \mathbb{Z}_{\ge 1} \cup \{+\infty\}$ is Borel. It is continuous if and only if $|m| = |n|$.

It now follows from Theorem 5.14 that the restriction of the partition (1.2) to the perfect kernel

$$ \mathcal{K}(\mathrm{BS}(m, n)) = \bigsqcup_{q \in \mathbb{Q}_{m,n}} \mathcal{K}_q(\mathrm{BS}(m, n)), \qquad (1.3) $$

where $\mathcal{K}_q(\mathrm{BS}(m, n)) := \mathcal{K}(\mathrm{BS}(m, n)) \cap \mathbf{Ph}^{-1}(q)$, satisfies all the conclusions of Theorem B. The pieces $\mathcal{K}_q(\mathrm{BS}(m, n))$ are indeed non-empty, see Remark 5.12.

## 1.3 Approximations by subgroups of other phenotypes

We still stick to the case $|m| \ne 1$ and $|n| \ne 1$. Since the only non-open piece in partition (1.2) is $\mathbf{Ph}^{-1}(\infty)$, the subgroups of infinite phenotype are the only ones which can be approximated in $\mathrm{Sub}(\mathrm{BS}(m, n))$ by subgroups of other (that is, finite) phenotypes.

The set of limits of subgroups of finite phenotype depends on whether we fix the phenotype or we let it vary. About approximations by subgroups with a constant phenotype, we have the following result (see Proposition 5.8 and Theorem 6.2).

**Theorem D.** Assume $|m|, |n| \ne 1$ and let us fix a finite $(m, n)$-phenotype $q$.

1. If $|m| = |n|$, then $\mathbf{Ph}^{-1}(q)$ is closed, hence no infinite phenotype subgroup can be approximated by subgroups of phenotype $q$.

2. If $|m| \ne |n|$, then an infinite phenotype subgroup $\Lambda$ can be approximated by subgroups of phenotype $q$ if and only if $\Lambda \le \langle\langle b \rangle\rangle$, where $\langle\langle b \rangle\rangle$ is the normal subgroup generated by $b$.
where $\mathcal{K}_q(\mathrm{BS}(m, n)) := \mathcal{K}(\mathrm{BS}(m, n)) \cap \mathbf{Ph}^{-1}(q)$, satisfies all the conclusions of Theorem B. The pieces $\mathcal{K}_q(\mathrm{BS}(m, n))$ are indeed non-empty, see Remark 5.12.

## 1.3 Approximations by subgroups of other phenotypes

We still stick to the case $|m| \neq 1$ and $|n| \neq 1$. Since the only non-open piece in partition (1.2) is $\mathbf{Ph}^{-1}(\infty)$, the subgroups of infinite phenotype are the only ones which can be approximated in $\mathrm{Sub}(\mathrm{BS}(m, n))$ by subgroups of other (that is, finite) phenotypes.

The set of limits of subgroups of finite phenotype depends on whether we fix the phenotype or we let it vary. About approximations by subgroups with a constant phenotype, we have the following result (see Proposition 5.8 and Theorem 6.2).

**Theorem D.** Assume $|m|, |n| \neq 1$ and let us fix a finite $(m, n)$-phenotype $q$.

1. If $|m| = |n|$, then $\mathbf{Ph}^{-1}(q)$ is closed, hence no infinite phenotype subgroup can be approximated by subgroups of phenotype $q$.

2. If $|m| \neq |n|$, then an infinite phenotype subgroup $\Lambda$ can be approximated by subgroups of phenotype $q$ if and only if $\Lambda \le \langle\langle b \rangle\rangle$, where $\langle\langle b \rangle\rangle$ is the normal subgroup generated by $b$.

It is remarkable that the set $\overline{\mathbf{Ph}^{-1}(q)} \cap \mathbf{Ph}^{-1}(\infty)$ is independent of $q$ in the previous result.

Allowing the finite phenotype to vary yields new limit points. Our result is the following (see Proposition 6.7 and Corollary 6.11).

**Theorem E.** Assume $|m|, |n| \neq 1$.

1. If $|m| = |n|$ then every infinite phenotype subgroup is a limit of finite (and varying) phenotypes subgroups.

2. On the contrary, if $|m| \neq |n|$, then the set of subgroups in $\mathbf{Ph}^{-1}(\infty)$ which are limits of finite (and varying) phenotypes subgroups has empty interior in $\mathbf{Ph}^{-1}(\infty)$.

Therefore, in the case $|m| = |n|$, all subgroups of infinite phenotype are limits of subgroups of finite phenotype, but none of them is a limit of subgroups of fixed finite phenotype.

The case $|m| \neq |n|$ is more complex. We do not have a nice description of the limit set from the above theorem. We can show however that this limit set
It is remarkable that the set $\overline{\mathbf{Ph}^{-1}(q)} \cap \mathbf{Ph}^{-1}(\infty)$ is independent of $q$ in the previous result.

Allowing the finite phenotype to vary yields new limit points. Our result is the following (see Proposition 6.7 and Corollary 6.11).

**Theorem E.** Assume $|m|, |n| \neq 1$.

1. If $|m| = |n|$ then every infinite phenotype subgroup is a limit of finite (and varying) phenotype subgroups.

2. On the contrary, if $|m| \neq |n|$, then the set of subgroups in $\mathbf{Ph}^{-1}(\infty)$ which are limits of finite (and varying) phenotypes subgroups has empty interior in $\mathbf{Ph}^{-1}(\infty)$.

Therefore, in the case $|m| = |n|$, all subgroups of infinite phenotype are limits of subgroups of finite phenotype, but none of them is a limit of subgroups of fixed finite phenotype.

The case $|m| \neq |n|$ is more complex. We do not have a nice description of the limit set from the above theorem. We can show however that this limit set is strictly larger than its fixed phenotype counterpart, see Proposition 6.12 and Theorem 6.14.

## 1.4 Closures of orbits in finite phenotype

We still stick to the case $|m| \neq 1$, $|n| \neq 1$, and assume moreover $|m| \neq |n|$. The previous subsection shows that for any finite phenotype $q$, we have

$$ \mathbf{Ph}^{-1}(q) \subsetneq \overline{\mathbf{Ph}^{-1}(q)} \subsetneq \mathbf{Ph}^{-1}(q) \cup \mathbf{Ph}^{-1}(\infty). $$

Theorem B further yields that $\mathbf{Ph}^{-1}(q)$ contains dense orbits. For such an orbit $O$, one has $\overline{O} = \overline{\mathbf{Ph}^{-1}(q)}$, thus $\overline{O}$ intersects $\mathbf{Ph}^{-1}(\infty)$. In fact, Theorem D completely described $\overline{O}$. We now turn our attention to the orbits whose closure is contained in $\mathbf{Ph}^{-1}(q)$. Quite remarkably, they form a compact set.

**Theorem F** (see Theorem 5.20). Suppose $|m|, |n| \neq 1$ and $|m| \neq |n|$. For every finite phenotype $q$, there is a positive integer $s = s(q, m, n)$ such that the subset

$$ \mathcal{MC}_q := \mathbf{Ph}^{-1}(q) \cap \{\Lambda \in \mathrm{Sub}(\mathrm{BS}(m, n)): \Lambda \geq \langle\langle b^s \rangle\rangle\} $$

is compact and contains all the invariant compact subsets of $\mathbf{Ph}^{-1}(q)$.
is strictly larger than its fixed phenotype counterpart, see Proposition 6.12 and Theorem 6.14.

## 1.4 Closures of orbits in finite phenotype

We still stick to the case $|m| \neq 1$, $|n| \neq 1$, and assume moreover $|m| \neq |n|$. The previous subsection shows that for any finite phenotype $q$, we have

$$ \mathbf{Ph}^{-1}(q) \subset \overline{\mathbf{Ph}^{-1}(q)} \subset \mathbf{Ph}^{-1}(q) \cup \mathbf{Ph}^{-1}(\infty). $$

Theorem B yields that $\mathbf{Ph}^{-1}(q)$ contains orbits that are unbounded (i.e. adherent to $\mathbf{Ph}^{-1}(\infty)$). In Theorem D, we described their limit points. We now turn our attention to the bounded orbits. Quite remarkably, they form a compact set.

**Theorem F** (see Theorem 5.20). Suppose $|m|, |n| \neq 1$ and $|m| \neq |n|$. For every finite phenotype $q$, there is a positive integer $s = s(q, m, n)$ such that the subset

$$ \mathcal{MC}_q := \mathbf{Ph}^{-1}(q) \cap \{\Lambda \in \mathrm{Sub}(\mathrm{BS}(m, n)): \Lambda \ge \langle\langle b^s \rangle\rangle\} $$

is compact and contains all the invariant compact subsets of $\mathbf{Ph}^{-1}(q)$.

In particular every normal subgroup of phenotype $q$, and hence every finite index subgroup, contains $\langle\langle b^s \rangle\rangle$. Moreover, $\mathcal{MC}_q \cap \mathcal{K}_q(\mathrm{BS}(m, n))$ has empty interior in $\mathcal{K}_q(\mathrm{BS}(m, n))$ (Theorem 5.20-(4)).

When $\mathrm{gcd}(m, n) = 1$, the above theorem takes an easier form: $s = q$ and $\mathcal{MC}_q \cap \mathcal{K}(\mathrm{BS}(m, n)) = \{\langle\langle b^q \rangle\rangle\}$. In particular, $\langle\langle b^q \rangle\rangle$ is the unique normal subgroup of phenotype $q$ and infinite index, see Theorem 5.20-(5). On the other hand, if $\mathrm{gcd}(m, n) \neq 1$, then the perfect kernel contains continuum many normal subgroups of phenotype $q$, see Theorem 5.24.

## 1.5 An example: the case of $\mathrm{BS}(2, 3)$

Let us specialize our theorems to the case of $\mathrm{BS}(2, 3)$. An illustrative picture is given in Figure 1.

Since $2 \neq 3$, Theorem A tells us that $\mathcal{K}(\mathrm{BS}(2, 3)) = \mathrm{Sub}_{[\infty]}(\mathrm{BS}(2, 3))$. In this case the phenotype is given by the following simple formula

$$ \mathbf{Ph}(\Lambda) = \frac{I}{2^{|I|_2}3^{|I|_3}} \quad \text{where } I := [\langle b \rangle : \Lambda \cap \langle b \rangle]. $$
In particular every normal subgroup of phenotype $q$, and hence every
finite index subgroup, contains $\langle\langle b^s \rangle\rangle$. Moreover, $\mathcal{MC}_q \cap \mathcal{K}_q(\text{BS}(m,n))$ has
empty interior in $\mathcal{K}_q(\text{BS}(m,n))$ (Theorem 5.20-(4)).

When $\gcd(m, n) = 1$, the above theorem takes an easier form: $s = q$ and $\mathcal{MC}_q \cap \mathcal{K}(\text{BS}(m, n)) = \{\langle\langle b^q \rangle\rangle\}$. In particular, $\langle\langle b^q \rangle\rangle$ is the unique normal subgroup of phenotype $q$ and infinite index, see Theorem 5.20-(5). On the other hand, if $\gcd(m, n) \neq 1$, then the perfect kernel contains continuum many normal subgroups of phenotype $q$, see Theorem 5.26.

1.5 An example: the case of $\text{BS}(2, 3)$

Let us specialize our theorems to the case of $\text{BS}(2, 3)$. An illustrative picture
is given in Figure 1.

<figure><img src="image_1.png" /><figcaption>Figure 1: The space of subgroups of $\text{BS}(2, 3)$</figcaption></figure>

Since $2 \neq 3$, Theorem A tells us that $\mathcal{K}(\text{BS}(2,3)) = \text{Sub}_{[\infty]}(\text{BS}(2,3))$. In
<figure><img src="image_2.png" /><figcaption>Figure 1: The space of subgroups of $BS(2, 3)$</figcaption></figure>

Therefore, the possible phenotypes for the subgroups of $BS(2, 3)$ are given by all the positive integers not divisible by 2 and 3, and infinity. Denoting $K_q = \{\Lambda \le BS(2, 3): Ph(\Lambda) = q\}$, the partition (1.3) becomes

$$ K(BS(2, 3)) = K_{\infty} \sqcup \bigsqcup_{q: \gcd(q,2)=\gcd(q,3)=1} K_q. $$

By Theorem B, the action on each $K_q$ is topologically transitive. Note that all finite index subgroups have finite phenotype. The set $K_{\infty}$ is closed and colored in black in Figure 1; the subsets $K_q$ are open and colored in gray in the figure. Finally the finite index subgroups are denoted by the dotted lines. Note that there are infinitely many finite index subgroups and they accumulate on the sets $K_q$.

Note that for every finite $q$, the set $\overline{K_q} \cap K_{\infty}$ is non-empty and independent of $q$; indeed by Theorem D this is the set of subgroups of infinite phenotype contained in $\langle\langle b \rangle\rangle$. This set is illustrated as the black central disk in the
this case the phenotype is given by the following simple formula

$$ \mathbf{Ph}(\Lambda) = \frac{I}{2^{|I|_2} 3^{|I|_3}}, $$

where $I$ is the index $I := [\langle b \rangle : \Lambda \cap \langle b \rangle]$, and where $|I|_p$ denotes the $p$-adic valuation of $I$ subject to the convention that $|\infty|_p = 0$.

Therefore, the possible phenotypes for the subgroups of $BS(2, 3)$ are given by all the positive integers not divisible by 2 and 3, and infinity. Denoting $\mathcal{K}_q = \{\Lambda \le BS(2, 3) : \mathbf{Ph}(\Lambda) = q\}$, the partition (1.3) becomes

$$ \mathcal{K}(BS(2, 3)) = \mathcal{K}_{\infty} \sqcup \bigcup_{q: \gcd(q,2)=\gcd(q,3)=1} \mathcal{K}_q. $$

By Theorem B, the action on each $\mathcal{K}_q$ is topologically transitive. Note that all finite index subgroups have finite phenotype. The set $\mathcal{K}_{\infty}$ is closed and colored in black in Figure 1; the subsets $\mathcal{K}_q$ are open and colored in gray in the figure. Finally the finite index subgroups are denoted by the dotted lines. Note that there are infinitely many finite index subgroups and they accumulate on the sets $\mathcal{K}_q$.

Note that for every finite $q$, the set $\overline{\mathcal{K}}_q \cap \mathcal{K}_{\infty}$ is non-empty and independent of $q$; indeed by Theorem D this is the set of subgroups of infinite phenotype contained in $\langle b \rangle$. This set is illustrated as the black central disk in the figure. As one can guess in the figure, $\cup_{q \text{ finite}} \overline{\mathcal{K}}_q \cap \mathcal{K}_{\infty}$ is strictly bigger than this set, and yet not the entirety of $\mathcal{K}_{\infty}$, as prescribed by Theorem E.

We finally apply Theorem F. Since $\gcd(2, 3) = 1$, for every finite phenotype $q$ the largest compact invariant subset of $\mathcal{K}_q$ consists only of one point: the unique normal subgroup contained in $\mathcal{K}_q$, namely $\langle b^q \rangle$, pictured with a small star in the figure. Moreover, $\mathcal{MC}_q$ consists of the finite index subgroups of phenotype $q$ represented by the dotted lines emanating from the star together with the single accumulation point $\langle b^q \rangle$ of $\mathcal{MC}_q$.

**Remark.** Figure 1 is actually quite general: as soon as $|m| \ne |n|$, we have the exact same picture except that the possible phenotypes are different, and the stars turn into bigger compact maximal invariant subsets. Moreover, the phenotype is given by a more complicated formula.

## 1.6 Some ideas on the techniques of proofs

The definition of the topology on the space of subgroups leads us to look at the restriction of transitive actions to some part of their Schreier graph and
figure. As one can guess in the figure, $\cup_q \text{finite}\mathcal{K}_q$ is strictly bigger than this
set, and yet not the entirety of $\mathcal{K}_\infty$, as prescribed by Theorem E.

We finally apply Theorem F. Since $\gcd(2,3) = 1$, for every finite pheno-
type $q$ the largest compact invariant subset of $\mathcal{K}_q$ consists only of one point:
the unique normal subgroup $\langle\langle b^q \rangle\rangle$ contained in $\mathcal{K}_q$, pictured with a star in
the figure.

**Remark.** Figure 1 is actually quite general: as soon as $|m| \neq |n|$, we have the exact same picture except that the possible phenotypes are different, and the stars turn into bigger compact maximal invariant subsets. Moreover, the phenotype is given by a more complicated formula.

1.6 Some ideas on the techniques of proofs

Topology questions lead us to look at the trace of transitive actions on some
parts of their Schreier graph and most statements consist in assembling such
parts from different actions (to form new actions): this leads us to the notion
of pre-action, as considered in [FMMS20], where to facilitate the verification
of the group relation, we impose that $b$ is defined everywhere, i.e. on the
whole domain of the pre-action (see Section 3.1). These pre-actions are
more malleable but the algebraic conditions underlying them still make them
difficult to manipulate.

We then move on to purely combinatorial objects associated with actions
and pre-actions: the (m, n)-graphs (Section 3.3). These are oriented graphs
which carry labels on the vertices and on the edges and which satisfy simple
arithmetic conditions linking valences and labels (Definition 3.12, equalities
(3.13) and inequalities (3.14)). They generalize the Bass-Serre graphs of pre-
actions used in [FMMS20] by adding their labels which record the size of the
orbits of b, b^m or b^n according to the graph element considered. Notice that
in [FMMS20] the b-orbits were assumed to be infinite.

All the vertex labels of a connected (m, n)-graph have the same (m, n)-
phenotype (Proposition 4.6) which is thus defined to be the phenotype of the
graph (Definition 4.8).

We have some gluing results between two (m, n)-graphs. The phenotype
is a complete invariant of gluing, more precisely: consider two connected
(m, n)-graphs that are non-saturated (at least one of the inequalities (3.14)
is strict); then they can appear as subgraphs of the same (m, n)-graph if and
only if they have the same phenotype (Theorem 4.13). This relies on the
then on assembling such parts from different actions (to form new actions):
this leads us to the notion of pre-action, as considered in [FLMMS22], where
to facilitate the verification of the group relation, we impose that $b$ is defined
everywhere, i.e. on the whole domain of the pre-action (see Section 3.1).
These pre-actions are more malleable but the algebraic conditions underlying
them still make them difficult to manipulate.

This is why we further downgrade the data and move on to purely com-
binatorial objects associated with actions and pre-actions: the (m,n)-graphs
(Section 3.3). These are oriented graphs which carry labels on the vertices
and on the edges and which satisfy simple arithmetic conditions linking de-
grees and labels (Definition 3.12, equalities (3.13) and inequalities (3.14)).
They generalize the Bass-Serre graphs of pre-actions used in [FLMMS22] by
adding their labels which record the size of the orbits of b, $b^m$ or $b^n$ according
to the graph element considered. Notice that in [FLMMS22] the b-orbits
were assumed to be infinite.

All the vertex labels of a connected (m, n)-graph have the same (m, n)-
phenotype (Proposition 4.6) which is thus defined to be the phenotype of the
graph (Definition 4.8).

At this level, we can consider assembling together different parts (orig-
inating from different actions). Consider two connected (m, n)-graphs that
are non-saturated (at least one of the inequalities (3.14) is strict), then they
can appear as subgraphs of the same (m, n)-graph as soon as they have the
same phenotype (Theorem 4.13). This relies on two basic constructions, the
Welding Lemma 4.16 and the Connecting Theorem 4.17.

We then proceed by upgrading from (m, n)-graphs to pre-actions and
actions (Proposition 3.23). These upgrades are not uniquely determined,
however if an (m, n)-graph $G_2$ contains the (m, n)-graph $G_1$ of a pre-action
$\alpha_1$, then the upgraded pre-action $\alpha_2$ can be chosen to extend $\alpha_1$ (Proposi-
tion 3.23).

To summarize, we will use several times the same construction scheme:
Considering two actions, we restrict them to a large but proper part of their
domain (pre-actions). We downgrade the resulting pre-actions to (m, n)-
graphs and glue them together. We saturate the resulting (m, n)-graph and
upgrade it into one action that "contains" the chosen parts of both original
actions as sub-pre-actions (Theorem 4.12).
Welding Lemma 4.16 and the Connecting Theorem 4.17.

We then have statements that allow us to upgrade $(m, n)$-graphs to pre-
actions. These upgrades are not univocal, however if an $(m, n)$-graph $\mathcal{G}_2$
contains the $(m, n)$-graph $\mathcal{G}_1$ of a pre-action $\alpha_1$, then the upgraded pre-action
$\alpha_2$ can be chosen to extend $\alpha_1$ (Proposition 3.23).

We will thus use several times the following construction scheme: consid-
ering two actions, we restrict them to their traces on large but proper parts
of their domain. We degrade the resulting pre-actions to (m, n)-graphs and
glue them together. We saturate the resulting (m, n)-graph and upgrade it
into an action which "contains" the traces of the original actions.

**Acknowledgements.** A. C. acknowledges funding by the Deutsche Forschungs-
gemeinschaft (DFG, German Research Foundation) - 281869850 (RTG 2229).
D. G. is supported by the CNRS and partially supported by the LABEX
MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program
"Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French Na-
tional Research Agency (ANR). F. L.M. acknowledges funding by the ANR
projects ANR-17-CE40-0026 AGRUME and ANR-19-CE40-0008 AODynG.

## 2 Preliminaries and notations

In this text, we denote by $\mathbb{Z}_{\ge 0} := \{0, 1, 2, \dots\}$ the set of non-negative integers
and by $\mathbb{Z}_{\ge 1} := \{1, 2, 3, \dots\}$ the set of positive integers. Given two integers
$k, l \in \mathbb{Z} \setminus \{0\}$, we denote by $\gcd(k, l) \in \mathbb{Z}_{\ge 1}$ the greatest common divisor of $k$
and $l$. We use the convention that $\gcd(k, \infty) = k$ and $\frac{\infty}{k} = k\infty = \infty$.

Let $\mathcal{P}$ be the set of prime numbers. Given an integer $k \in \mathbb{Z}\setminus\{0\}$ and a
prime $p \in \mathcal{P}$, we denote by $|k|_p$ the $p$-adic valuation of $k$, that is $|k|_p$ is the
largest positive integer such that $p^{|k|_p}$ divides $k$.

## 2.1 Graphs and Schreier graphs

All our graphs are defined as in [Ser80]. That is, a graph $\mathcal{G}$ is a couple
$(V(\mathcal{G}), E(\mathcal{G}))$ where $V(\mathcal{G})$ is the **vertex set** and $E(\mathcal{G})$ is the **edge set**, en-
dowed with:

• two maps $s, t: E(\mathcal{G}) \to V(\mathcal{G})$ called **source** and **target** respectively;

• a fixed-point-free involution $E(\mathcal{G}) \to E(\mathcal{G}), e \mapsto \bar{e}$;
## 1.7 Subsequent work

Since the first version of the present paper appeared, two preprints have
enriched the picture as follows.

On the one hand, the three last-named authors proved in [GLMS24]
that the dynamics on the pieces $\mathcal{K}_q$ is in fact highly topologically transitive.
They also studied the property of high transitivity for transitive actions of
$\text{BS}(m, n)$: they characterized the pieces containing subgroups $\Lambda$ such that
the action $\Lambda \setminus \text{BS}(m, n) \to \text{BS}(m, n)$ is highly transitive and they established
that this property is generic in these pieces.

On the other hand, Sasha Bontemps has extended Theorems A, B and C
to generalized Baumslag-Solitar groups, where the right notion of phenotype
is more subtle [Bon24]. She also obtained high topological transitivity results
generalizing Theorem C from the aforementioned preprint [GLMS24].

**Acknowledgements.** We are very grateful to both referees for their work and their detailed remarks which helped us to improve the paper. A. C. acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 281869850 (RTG 2229). D. G. is supported by the CNRS and partially supported by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program “Investissements d’Avenir” (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). F. L.M. acknowledges funding by the ANR projects ANR-17-CE40-0026 AGRUME and ANR-19-CE40-0008 AODynG.

# 2 Preliminaries and notations

In this text, we denote by $\mathbb{Z}_{\ge 0} := \{0, 1, 2, \dots\}$ the set of non-negative integers
and by $\mathbb{Z}_{\ge 1} := \{1, 2, 3, \dots\}$ the set of positive integers. Given two integers
$k, l \in \mathbb{Z} \setminus \{0\}$, we denote by $\gcd(k, l) \in \mathbb{Z}_{\ge 1}$ the greatest common divisor of $k$
and $l$. We use the convention that $\gcd(k, \infty) = k$ and $\frac{\infty}{k} = k\infty = \infty$.

Let $\mathcal{P}$ be the set of prime numbers. Given an integer $k \in \mathbb{Z} \setminus \{0\}$ and a
prime $p \in \mathcal{P}$, we denote by $|k|_p$ the p-adic valuation of $k$, that is $|k|_p$ is the
largest positive integer such that $p^{|k|_p}$ divides $k$.
such that $\mathbf{s}(\bar{e}) = \mathbf{t}(e)$ and $\mathbf{t}(\bar{e}) = \mathbf{s}(e)$.

An **orientation** of the graph $\mathcal{G}$ is a partition $E(\mathcal{G}) = E^+(\mathcal{G}) \sqcup E^-(\mathcal{G})$ whose pieces are exchanged by the involution $e \mapsto \bar{e}$. Edges in $E^+(\mathcal{G})$ are called **positive** edges and edges in $E^-(\mathcal{G})$ are **negative**.

**Remark 2.1.** In order to define an oriented graph $\mathcal{G}$, it is enough to define the set of vertices $V(\mathcal{G})$, the set of positive edges $E^+(\mathcal{G})$, and the restrictions of the source and target maps $\mathbf{s}, \mathbf{t}$ to $E^+(\mathcal{G})$. Indeed, we can define $E^-(\mathcal{G})$ to be a copy of $E^+(\mathcal{G})$ and the involution $e \mapsto \bar{e}$ to be the natural identification of $E^+(\mathcal{G})$ with $E^-(\mathcal{G})$. We extend the source and target map by setting $\mathbf{s}(\bar{e}) := \mathbf{t}(e)$ and $\mathbf{t}(\bar{e}) := \mathbf{s}(e)$.

The **degree** a vertex $v$ in a graph $\mathcal{G}$, is the cardinal

$$ \deg(v) := |\{e \in E(\mathcal{G}) : \mathbf{s}(e) = v\}| = |\{e \in E(\mathcal{G}) : \mathbf{t}(e) = v\}|. $$

If $\mathcal{G}$ is oriented, we say that an edge $e$ is:

* a **v-outgoing** edge if it is positive and $\mathbf{s}(e) = v$;

* a **v-incoming** edge if it is positive and $\mathbf{t}(e) = v$.

The **outgoing degree** $\deg_{\text{out}}(v)$ of $v$ is the number of $v$-outgoing edges while its **incoming degree** $\deg_{\text{in}}(v)$ is the number of $v$-incoming edges. We clearly have $\deg_{\text{out}}(v) + \deg_{\text{in}}(v) = \deg(v)$.

A **subgraph** $\mathcal{G}'$ of a graph $\mathcal{G}$ is a graph such that $V(\mathcal{G}') \subseteq V(\mathcal{G})$, $E(\mathcal{G}') \subseteq E(\mathcal{G})$ and the structural maps of $\mathcal{G}'$ are restrictions of those of $\mathcal{G}$.

Still following [Ser80], we call **circuit** a subgraph isomorphic to a circular graph (of length $l \ge 1$) and **loop** a circuit of length 1. We also call loop an edge such that $\mathbf{s}(e) = \mathbf{t}(e)$. A **path** in a graph $\mathcal{G}$ is a finite sequence of edges $(e_1, \dots, e_n)$, such that for all $1 \le k \le n-1$, $\mathbf{t}(e_k) = \mathbf{s}(e_{k+1})$. Similarly, an **infinite path** is a sequence of edges $(e_k)_{k \ge 1}$ such that $\mathbf{t}(e_k) = \mathbf{s}(e_{k+1})$ for all $k \ge 1$. Finally a (possibly infinite) path is called **simple** when the induced sequence of vertices is injective.

The **ball** $B(v, R)$ of radius $R$ centered at a vertex $v$ in a graph $\mathcal{G}$ is the subgraph induced by the set of vertices of $\mathcal{G}$ at distance $\le R$ from $v$ in the path metric.

**Schreier graphs** Let $\Gamma$ be a group and let $S$ be a generating set of $\Gamma$. Consider a (right) action $\alpha: X \hookrightarrow \Gamma$. The **Schreier graph** of $\alpha$ relatively to $S$ is the oriented graph $\mathbf{Sch}(\alpha) = \mathbf{Sch}(\alpha, S)$ defined by

$$ V(\mathbf{Sch}(\alpha)) := X \text{ and } E^+(\mathbf{Sch}(\alpha)) := \{(x, s) : x \in X, s \in S\} $$
## 2.1 Graphs and Schreier graphs

All our graphs are defined as in [Ser80]. That is, a graph $\mathcal{G}$ is a couple $(V(\mathcal{G}), E(\mathcal{G}))$ where $V(\mathcal{G})$ is the **vertex set** and $E(\mathcal{G})$ is the **edge set**, endowed with:

* two maps $s, t: E(\mathcal{G}) \to V(\mathcal{G})$ called **source** and **target** respectively;

* a fixed-point-free involution $E(\mathcal{G}) \to E(\mathcal{G})$, $e \mapsto \bar{e}$;

such that $s(\bar{e}) = t(e)$ and $t(\bar{e}) = s(e)$.

An **orientation** of the graph $\mathcal{G}$ is a partition $E(\mathcal{G}) = E^+(\mathcal{G}) \sqcup E^-(\mathcal{G})$ whose pieces are exchanged by the involution $e \mapsto \bar{e}$. Edges in $E^+(\mathcal{G})$ are called **positive edges** and edges in $E^-(\mathcal{G})$ are **negative**.

**Remark 2.1.** In order to define an oriented graph $\mathcal{G}$, it is enough to define the set of vertices $V(\mathcal{G})$, the set of positive edges $E^+(\mathcal{G})$, and the restrictions of the source and target maps $s, t$ to $E^+(\mathcal{G})$. Indeed, we can define $E^-(\mathcal{G})$ to be a copy of $E^+(\mathcal{G})$ and the involution $e \mapsto \bar{e}$ to be the natural identification of $E^+(\mathcal{G})$ with $E^-(\mathcal{G})$. We extend the source and target map by setting $s(\bar{e}) := t(e)$ and $t(\bar{e}) := s(e)$.

The **degree** of a vertex $v$ in a graph $\mathcal{G}$, is the cardinal

$$ \deg(v) := |\{e \in E(\mathcal{G}) : s(e) = v\}| = |\{e \in E(\mathcal{G}) : t(e) = v\}|. $$

If $\mathcal{G}$ is oriented, we say that an edge $e$ is:

* a **v-outgoing** edge if it is positive and $s(e) = v$;

* a **v-incoming** edge if it is positive and $t(e) = v$.

The **outgoing degree** $\deg_{\text{out}}(v)$ of $v$ is the number of $v$-outgoing edges while its **incoming degree** $\deg_{\text{in}}(v)$ is the number of $v$-incoming edges. We clearly have $\deg_{\text{out}}(v) + \deg_{\text{in}}(v) = \deg(v)$.

A **subgraph** $\mathcal{G}'$ of a graph $\mathcal{G}$ is a graph such that $V(\mathcal{G}') \subseteq V(\mathcal{G})$, $E(\mathcal{G}') \subseteq E(\mathcal{G})$ and the structural maps of $\mathcal{G}'$ are restrictions of those of $\mathcal{G}$.

Still following [Ser80], we call **circuit** a subgraph isomorphic to a circular graph (of length $l \ge 1$) and **loop** a circuit of length 1. The edge of a loop is also called a loop.

A **path** in a graph $\mathcal{G}$ is a finite sequence of edges $(e_1, \dots, e_n)$, such that for all $1 \le k \le n-1$, $t(e_k) = s(e_{k+1})$. Similarly, an **infinite path** is a sequence of edges $(e_k)_{k \ge 1}$ such that $t(e_k) = s(e_{k+1})$ for all $k \ge 1$. Finally a (possibly infinite) path is called **simple** when the induced sequence of vertices is injective.
where $\mathfrak{s}(x, s) = x$ and $\mathfrak{t}(x, s) = xs$, together with the following labeling: the edge $(x, s)$ is labeled $s$ and its opposite $(\overline{x}, s)$ is labeled by $s^{-1}$.

Given a subgroup $\Lambda \leq \Gamma$, we denote by $\mathbf{Sch}(\Lambda, S)$ the Schreier graph of the natural action $\Lambda \setminus \Gamma \hookrightarrow \Gamma$.

The **Cayley graph** of $\Gamma$ relatively to $S$ is the Schreier graph $\mathbf{Sch}(\alpha, S)$ of the action $\alpha: \Gamma \hookrightarrow \Gamma$ by (right) translations. This graph is denoted by $\mathbf{Cay}(\Gamma, S)$ and clearly $\mathbf{Cay}(\Gamma, S) = \mathbf{Sch}(\{\mathrm{id}\}, S)$. The $\Gamma$-action by left translations extends to the standard left action of $\Gamma$ on $\mathbf{Cay}(\Gamma, S)$ by graph automorphisms $^1$. In particular $\Lambda \setminus \mathbf{Cay}(\Gamma, S) = \mathbf{Sch}(\Lambda, S)$.

Let $\varphi: X \to Y$ be a $\Gamma$-equivariant map from $\alpha: X \hookrightarrow \Gamma$ to $\beta: Y \hookrightarrow \Gamma$ and let $S$ be a generating set of $\Gamma$. The map $\varphi$ extends to a graph morphism from $\mathbf{Sch}(\alpha, S)$ to $\mathbf{Sch}(\beta, S)$ which respects the labelings. In particular, given subgroups $\Lambda_1 \leq \Lambda_2 \leq \Gamma$, the equivariant map $\Lambda_1 \setminus \Gamma \to \Lambda_2 \setminus \Gamma$ defines a surjective morphism $\mathbf{Sch}(\Lambda_1, S) \to \mathbf{Sch}(\Lambda_2, S)$.

## 2.2 Space of subgroups

Let $\Gamma$ be a countable group. We identify its set of subsets with $\{0, 1\}^{\Gamma}$ and we endow it with the product topology, thus turning it into a Polish compact space. The **space of subgroups** of $\Gamma$ is the closed, hence compact Polish, subspace

$$ \mathrm{Sub}(\Gamma) := \{\Lambda \in \{0, 1\}^{\Gamma} : \Lambda \text{ is a subgroup}\}, $$

which is also totally disconnected. The clopen subsets

$$ \mathcal{V}(I, O) := \{\Lambda \in \mathrm{Sub}(\Gamma) : I \subseteq \Lambda \text{ and } \Lambda \cap O = \emptyset\} $$

of $\mathrm{Sub}(\Gamma)$ where $I, O$ run over finite subsets of $\Gamma$, form a basis of the topology. Note that a sequence $(\Lambda_n)_{n \geq 0}$ of subgroups converges to $\Lambda$ if and only if for all $\gamma \in \Gamma$,

$$ (\gamma \in \Lambda) \iff (\gamma \in \Lambda_i \text{ for } i \text{ large enough}). $$

By the Cantor-Bendixson Theorem [Can1884, Ben1883] (see e.g. [Kec95, Thm. 6.4]), there is a unique decomposition

$$ \mathrm{Sub}(\Gamma) = \mathcal{C}(\Gamma) \sqcup \mathcal{K}(\Gamma) $$

where $\mathcal{C}(\Gamma)$ is a countable open subset and $\mathcal{K}(\Gamma)$ is a closed perfect$^2$ subspace called the **perfect kernel** of $\Gamma$. The set $\mathcal{K}(\Gamma)$ is the largest subset

<footnote>$^1$This is why Schreier graphs were defined with respect to right actions.</footnote>

<footnote>$^2$A topological space is called **perfect** if it has no isolated points.</footnote>
The **ball** $B(v, R)$ of radius $R$ centered at a vertex $v$ in a graph $\mathcal{G}$ is the subgraph induced by the set of vertices of $\mathcal{G}$ at distance $\le R$ from $v$ in the path metric.

**Schreier graphs.** Let $\Gamma$ be a group and let $S$ be a generating set of $\Gamma$. Consider a (right) action $\alpha: X \ltimes \Gamma$. The **Schreier graph** of $\alpha$ relatively to $S$ is the oriented graph $\mathbf{Sch}(\alpha) = \mathbf{Sch}(\alpha, S)$ defined by

$$V(\mathbf{Sch}(\alpha)) := X \text{ and } E^{+}(\mathbf{Sch}(\alpha)) := \{(x, s) : x \in X, s \in S\}$$

where $s(x, s) = x$ and $t(x, s) = xs$, together with the following labeling: the edge $(x, s)$ is labeled $s$ and its opposite $(\overline{x, s})$ is labeled by $s^{-1}$.

Given a subgroup $\Lambda \le \Gamma$, we denote by $\mathbf{Sch}(\Lambda, S)$ the Schreier graph of the natural action $\Lambda \ltimes \Gamma \to \Gamma$.

The **Cayley graph** of $\Gamma$ relatively to $S$ is the Schreier graph $\mathbf{Sch}(\alpha, S)$ of the action $\alpha: \Gamma \ltimes \Gamma$ by (right) translations. This graph is denoted by $\mathbf{Cay}(\Gamma, S)$ and clearly $\mathbf{Cay}(\Gamma, S) = \mathbf{Sch}(\{\mathrm{id}, S\})$. The $\Gamma$-action by left translations extends to the standard left action of $\Gamma$ on $\mathbf{Cay}(\Gamma, S)$ by graph automorphisms $^1$. In particular $\Lambda \ltimes \mathbf{Cay}(\Gamma, S) = \mathbf{Sch}(\Lambda, S)$.

Let $\varphi: X \to Y$ be a $\Gamma$-equivariant map from $\alpha: X \ltimes \Gamma$ to $\beta: Y \ltimes \Gamma$ and let $S$ be a generating set of $\Gamma$. The map $\varphi$ extends to a graph morphism from $\mathbf{Sch}(\alpha, S)$ to $\mathbf{Sch}(\beta, S)$ which respects the labelings. In particular, given subgroups $\Lambda_1 \le \Lambda_2 \le \Gamma$, the equivariant map $\Lambda_1 \ltimes \Gamma \to \Lambda_2 \ltimes \Gamma$ defines a surjective morphism $\mathbf{Sch}(\Lambda_1, S) \to \mathbf{Sch}(\Lambda_2, S)$.

## 2.2 Space of subgroups

Let $\Gamma$ be a countable group. We identify its set of subsets with $\{0, 1\}^{\Gamma}$ and we endow it with the product topology, thus turning it into a Polish compact space. The **space of subgroups** of $\Gamma$ is the closed, hence compact Polish, subspace

$$\mathrm{Sub}(\Gamma) := \{\Lambda \in \{0, 1\}^{\Gamma} : \Lambda \text{ is a subgroup}\},$$

which is also totally disconnected. The clopen subsets

$$\mathcal{V}(I, O) := \{\Lambda \in \mathrm{Sub}(\Gamma) : I \subseteq \Lambda \text{ and } \Lambda \cap O = \emptyset\}$$

<footnote n="1">¹This is why Schreier graphs were defined with respect to right actions.</footnote>
$\mathcal{K} \subseteq \text{Sub}(\Gamma)$ without isolated points for the induced topology. In fact, $\mathcal{K}(\Gamma)$ is exactly the set of **condensation points**, that is, the points whose neighborhoods in $\text{Sub}(\Gamma)$ are all uncountable.

**Remark 2.2.** By a theorem of Brouwer, the space $\mathcal{K}(\Gamma)$ is either empty or a Cantor space, see [Kec95, Thm. 7.4].

**Remark 2.3.** The subset $\text{Sub}_{[\infty]}$ of infinite index subgroups of $\Gamma$ is closed in $\Gamma$ if and only if $\Gamma$ is finitely generated. Indeed if $\Gamma$ is finitely generated, then its finite index subgroups are isolated. If $\Gamma$ is not finitely generated, then all its finite index subgroups are not finitely generated. Such a subgroup is a limit of finitely generated subgroups, thus of infinite index.

The group $\Gamma$ acts (on the right) by conjugation via $\Lambda \cdot \gamma := \gamma^{-1}\Lambda\gamma$ on the space of its subgroups $\text{Sub}(\Gamma)$. This action is continuous and the Cantor-Bendixson decomposition $\text{Sub}(\Gamma) = \mathcal{C}(\Gamma) \sqcup \mathcal{K}(\Gamma)$ is $\Gamma$-invariant.

By the Baire category theorem, any countable closed subset of $\text{Sub}(\Gamma)$ contains an isolated point, so $\text{Sub}(\Gamma)$ has trivial perfect kernel if and only if it is countable. The following well-known proposition is useful for showing the latter property.

**Proposition 2.4.** Let $\Gamma$ be a countable group, let $N$ be a normal subgroup of $\Gamma$ such that $\Gamma/N$ is Noetherian (all its subgroups are finitely generated), and assume that $\text{Sub}(N)$ is countable. Then $\text{Sub}(\Gamma)$ is countable.

*Proof.* Let $\Lambda \leq \Gamma$ and denote by $\pi: \Gamma \to \Gamma/N$ the quotient map. Since $\Gamma/N$ is Noetherian, we have $\pi(\Lambda) = \langle S \rangle$ for some finite set $S$. Fix a finite set $S' \subseteq \Lambda$ such that $\pi(S') = S$. Then we can recover $\Lambda$ from $S'$ and its intersection with $N$ as

$$ \Lambda = \langle S' \cup (\Lambda \cap N) \rangle. $$

In other words, the map $(S', N') \mapsto \langle S' \cup N' \rangle$ surjects $\mathcal{P}_f(\Gamma) \times \text{Sub}(N)$ onto $\text{Sub}(\Gamma)$, where $\mathcal{P}_f(\Gamma)$ is the set of finite subsets of $\Gamma$, which is countable. Since $\text{Sub}(N)$ is countable as well we conclude that $\text{Sub}(\Gamma)$ is countable. $\square$

**Corollary 2.5.** If $|m| = 1$ or $|n| = 1$ then $\text{Sub}(\text{BS}(m, n))$ is countable.

*Sketch of proof.* We sketch the proof contained in [BLT19, Cor. 8.4]. By symmetry we may as well assume $m = 1$. Then $\text{BS}(m, n)$ is isomorphic to the semi-direct product $\mathbb{Z}[1/n] \times \mathbb{Z}$ where $\mathbb{Z}$ acts by multiplication by $n$. As explained in the proof of [BLT19, Cor. 8.4], $\text{Sub}(\mathbb{Z}[1/n])$ is countable, so the result follows from the previous proposition. $\square$
of Sub(Γ) where I, O run over finite subsets of Γ, form a basis of the topology. Note that a sequence $(\Lambda_n)_{n \ge 0}$ of subgroups converges to Λ if and only if for all $\gamma \in \Gamma$,

$$ (\gamma \in \Lambda) \iff (\gamma \in \Lambda_i \text{ for } i \text{ large enough}). $$

By the Cantor-Bendixson Theorem [Can84, Ben83] (see e.g. [Kec95, Thm. 6.4]), there is a unique decomposition

$$ \mathrm{Sub}(\Gamma) = \mathcal{C}(\Gamma) \sqcup \mathcal{K}(\Gamma) $$

where $\mathcal{C}(\Gamma)$ is a countable open subset and $\mathcal{K}(\Gamma)$ is a closed perfect<fnref n="2" /> subspace called the **perfect kernel** of $\Gamma$. The set $\mathcal{K}(\Gamma)$ is the largest subset $\mathcal{K} \subseteq \mathrm{Sub}(\Gamma)$ without isolated points for the induced topology. In fact, $\mathcal{K}(\Gamma)$ is exactly the set of **condensation points**, that is, the points whose neighborhoods in $\mathrm{Sub}(\Gamma)$ are all uncountable.

**Remark 2.2.** By a theorem of Brouwer, the space $\mathcal{K}(\Gamma)$ is either empty or a Cantor space, see [Kec95, Thm. 7.4].

**Remark 2.3.** The subset $\mathrm{Sub}_{[\infty]}(\Gamma)$ of infinite index subgroups of $\Gamma$ is closed in $\Gamma$ if and only if $\Gamma$ is finitely generated. Indeed if $\Gamma$ is finitely generated, then its finite index subgroups are isolated. If $\Gamma$ is not finitely generated, its finite index subgroups are not finitely generated, but they are limit points of finitely generated (thus of infinite index) subgroups; so $\mathrm{Sub}_{[\infty]}(\Gamma)$ is dense in $\mathrm{Sub}(\Gamma)$.

The group $\Gamma$ acts (on the right) by conjugation via $\Lambda \cdot \gamma := \gamma^{-1}\Lambda\gamma$ on the space of its subgroups $\mathrm{Sub}(\Gamma)$. This action is continuous and the Cantor-Bendixson decomposition $\mathrm{Sub}(\Gamma) = \mathcal{C}(\Gamma) \sqcup \mathcal{K}(\Gamma)$ is $\Gamma$-invariant.

By the Baire category theorem, any countable closed subset of $\mathrm{Sub}(\Gamma)$ contains an isolated point, so $\mathrm{Sub}(\Gamma)$ has trivial perfect kernel if and only if it is countable. The following well-known proposition is useful for showing the latter property.

**Proposition 2.4.** Let $\Gamma$ be a countable group, let $N$ be a normal subgroup of $\Gamma$ such that $\Gamma/N$ is Noetherian (all its subgroups are finitely generated), and assume that $\mathrm{Sub}(N)$ is countable. Then $\mathrm{Sub}(\Gamma)$ is countable.

<footnote n="2">²A topological space is called **perfect** if it has no isolated points.</footnote>
**Space of pointed actions** Let us now interpret the topological space Sub(Γ) in terms of pointed transitive group actions and their pointed Schreier graphs. To any pointed transitive group action (α, v), where α : V ↪- Γ and v ∈ V, we associate the stabilizer Stab_α(v) ∈ Sub(Γ), and we notice that Stab_α₁(v₁) = Stab_α₂(v₂) if and only if (α₁, v₁) and (α₂, v₂) are isomorphic as pointed transitive actions.

**Notation 2.6.** We denote by [α, v] the isomorphism class of any pointed transitive action (α, v).

We therefore have a canonical bijection [α, v] ↦ Stab_α(v) between the collection of isomorphism classes of pointed transitive actions and Sub(Γ). Its inverse is given by Λ ↦ [Λ \ Γ ↪- Γ, Λ]. Through this bijection, the action by conjugation of Γ on Sub(Γ) becomes [α, v] · γ = [α, vα(γ)], i.e., it moves the basepoint.

Via the above identification, we obtain a topology on the set of isomorphism classes of pointed actions [α, v].

It is clear that two pointed actions are isomorphic if and only if their Schreier graphs are isomorphic as pointed labeled graphs. Given two pointed labeled oriented graphs (G, v), (H, w) and a positive integer R, we write (G, v) ≃_R (H, w) to mean that the R-balls around v in G and around w in H are isomorphic as pointed oriented labeled graphs. It is an exercise to check that if Γ is generated by a finite set S, then the sets of the form

$$
\mathcal{N}([\alpha, v], R) := \{[\alpha', v'] : (\mathbf{Sch}(\alpha, S), v) \simeq_R (\mathbf{Sch}(\alpha', S), v')\}, \quad (2.7)
$$

constitute a basis of clopen neighborhoods of [α, v].

2.3 Bass-Serre theory

Associated with the standard HNN-presentation of

$$
\mathrm{BS}(m, n) = \langle b, t | tb^m t^{-1} = b^n \rangle,
$$

we have the BS(m, n)-action on its Bass-Serre tree T. Recall that T is the oriented tree with V(T) = BS(m, n)/⟨b⟩, E⁺(T) = BS(m, n)/⟨bⁿ⟩,

$$
\mathfrak{s}(\gamma \langle b^n \rangle) = \gamma \langle b \rangle, \text{ and } \mathfrak{t}(\gamma \langle b^n \rangle) = \gamma t \langle b \rangle
$$

and given a subgroup Λ ≤ BS(m, n), the quotient Λ \ T has the structure of
a graph of groups whose fundamental group is Λ, see [Ser80].
*Proof.* Let $\Lambda \leq \Gamma$ and denote by $\pi: \Gamma \to \Gamma/N$ the quotient map. Since $\Gamma/N$ is Noetherian, we have $\pi(\Lambda) = \langle S \rangle$ for some finite set $S$. Fix a finite set $S' \subseteq \Lambda$ such that $\pi(S') = S$. Then we can recover $\Lambda$ from $S'$ and its intersection with $N$ as

$$ \Lambda = \langle S' \cup (\Lambda \cap N) \rangle. $$

In other words, the map $(S', N') \mapsto \langle S' \cup N' \rangle$ surjects $\mathcal{P}_f(\Gamma) \times \text{Sub}(N)$ onto $\text{Sub}(\Gamma)$, where $\mathcal{P}_f(\Gamma)$ is the set of finite subsets of $\Gamma$, which is countable. Since $\text{Sub}(N)$ is countable as well we conclude that $\text{Sub}(\Gamma)$ is countable. $\square$

**Corollary 2.5.** If $|m| = 1$ or $|n| = 1$ then $\text{Sub}(\text{BS}(m, n))$ is countable.

*Sketch of proof.* We sketch the proof contained in [BLT19, Cor. 8.4]. By symmetry we may as well assume $m = 1$. Then $\text{BS}(m, n)$ is isomorphic to the semi-direct product $\mathbb{Z}[1/n] \times \mathbb{Z}$ where $\mathbb{Z}$ acts by multiplication by $n$. As explained in the proof of [BLT19, Cor. 8.4], $\text{Sub}(\mathbb{Z}[1/n])$ is countable, so the result follows from the previous proposition. $\square$

**Space of pointed actions.** Let us now interpret the topological space $\text{Sub}(\Gamma)$ in terms of pointed transitive group actions and their pointed Schreier graphs. To any pointed transitive group action $(\alpha, v)$, where $\alpha: V \hookrightarrow \Gamma$ and $v \in V$, we associate the stabilizer $\text{Stab}_\alpha(v) \in \text{Sub}(\Gamma)$, and we notice that $\text{Stab}_{\alpha_1}(v_1) = \text{Stab}_{\alpha_2}(v_2)$ if and only if $(\alpha_1, v_1)$ and $(\alpha_2, v_2)$ are isomorphic as pointed transitive actions.

**Notation 2.6.** We denote by $[\alpha, v]$ the isomorphism class of any pointed transitive action $(\alpha, v)$.

We therefore have a canonical bijection $[\alpha, v] \mapsto \text{Stab}_\alpha(v)$ between the collection of isomorphism classes of pointed transitive actions and $\text{Sub}(\Gamma)$. Its inverse is given by $\Lambda \mapsto [\Lambda \setminus \Gamma \hookrightarrow \Gamma, \Lambda]$. Through this bijection, the action by conjugation of $\Gamma$ on $\text{Sub}(\Gamma)$ becomes $[\alpha, v] \cdot \gamma = [\alpha, v\alpha(\gamma)]$, i.e., it moves the basepoint.

Via the above identification, we obtain a topology on the set of isomorphism classes of pointed actions $[\alpha, v]$.

It is clear that two pointed actions are isomorphic if and only if their Schreier graphs are isomorphic as pointed labeled graphs. Given two pointed labeled oriented graphs $(\mathcal{G}, v)$, $(\mathcal{H}, w)$ and a positive integer $R$, we write $(\mathcal{G}, v) \simeq_R (\mathcal{H}, w)$ to mean that the $R$-balls around $v$ in $\mathcal{G}$ and around $w$
**Remark 2.8.** Let $\Lambda \leqslant \mathrm{BS}(m, n)$ be a subgroup. If $\Lambda \cap \langle b \rangle = \{\mathrm{id}\}$, then $\Lambda$ acts freely on $\mathcal{T}$; thus it is the fundamental group of the quotient graph $\Lambda \backslash \mathcal{T}$, hence $\Lambda$ is a free group.

Let us now concentrate on a subgroup $\Lambda \leqslant \mathrm{BS}(m, n)$ such that $\Lambda \cap \langle b \rangle \neq \{\mathrm{id}\}$. Then for the induced action $\Lambda \rightsquigarrow \mathcal{T}$, each edge and vertex stabilizer is infinite cyclic: the tree $\mathcal{T}$ is a GBS-tree (for Generalized Baumslag-Solitar), in the sense of [For07, Lev07]. One can use this point of view to understand the graph of groups description of $\Lambda$. However, taking advantage of the transitivity of the $\mathrm{BS}(m, n)$-action on the edges and the vertices, we provide a slightly more precise description.

**Proposition 2.9.** Let $m$ and $n$ be non-zero integers. Let $\Lambda \leqslant \mathrm{BS}(m, n)$ be a subgroup such that $\Lambda \cap \langle b \rangle \neq \{\mathrm{id}\}$. The quotient graph of groups arising from the action $\Lambda \rightsquigarrow \mathcal{T}$ is isomorphic to the graph of groups obtained by attaching a copy of $\mathbb{Z}$ to every vertex and every edge of the quotient graph $\Lambda \backslash \mathcal{T}$, with structural maps of positive edges

$$ \begin{aligned} \mathbb{Z}_e &\to \mathbb{Z}_{\mathfrak{s}(e)}, & k &\mapsto \frac{n}{\mathrm{deg}_{\mathrm{out}}(\mathfrak{s}(e))} \cdot k, \\ \mathbb{Z}_e &\to \mathbb{Z}_{\mathfrak{t}(e)}, & k &\mapsto \frac{m}{\mathrm{deg}_{\mathrm{in}}(\mathfrak{t}(e))} \cdot k. \end{aligned} $$

*Proof.* In this proof we set $\Gamma := \mathrm{BS}(m, n)$. Let us consider the action of $\Lambda$ on the tree $\mathcal{T}$. Since $\mathcal{T}$ is locally finite, any edge adjacent to a vertex with infinite stabilizer has itself infinite stabilizer. It follows that all vertex and edge $\Lambda$-stabilizers are infinite. Being subgroups of the $\Gamma$-stabilizers, they are all isomorphic to $\mathbb{Z}$.

Observe that since $\Gamma$ acts transitively and the $\Gamma$-stabilizers are abelian, the $\Gamma$-stabilizers are canonically pairwise isomorphic: given any vertex $u \in V(\mathcal{T})$ and $a \in \mathrm{Stab}_{\Gamma}(u)$, one has

$$ \mathrm{gag}^{-1} = \mathrm{hah}^{-1} \quad \text{for any } g, h \in \Gamma \text{ such that } gu = hu. \qquad (2.10) $$

Indeed since $h^{-1}g \in \mathrm{Stab}_{\Gamma}(u)$, we get that $h^{-1}\mathrm{gag}^{-1}h = a$.

We now focus on the quotient graph of groups arising from the action $\Lambda \rightsquigarrow \mathcal{T}$. Let us recall from [Ser80] that its vertex groups are $G_v := \mathrm{Stab}_{\Lambda}(\tilde{v})$ and edge groups are $G_e := \mathrm{Stab}_{\Lambda}(\tilde{e})$, where $\tilde{v}, \tilde{e}$ are some lifts of $v, e$ in $\mathcal{T}$. Given any $e \in E^{+}(\Lambda \backslash \mathcal{T})$, the structural map $G_e \to G_{\mathfrak{t}(e)}$ is

$$ \begin{array}{ccc} G_e = \mathrm{Stab}_{\Lambda}(\tilde{e}) & \hookrightarrow & \mathrm{Stab}_{\Lambda}(\mathfrak{t}(\tilde{e})) \to \mathrm{Stab}_{\Lambda}(\widetilde{\mathfrak{t}(e)}) = G_{\mathfrak{t}(e)} \\ a & \mapsto & a & \mapsto \\ & & \mathrm{gag}^{-1} & \end{array} \qquad (2.11) $$
in $\mathcal{H}$ are isomorphic as pointed oriented labeled graphs. It is an exercise to check that if $\Gamma$ is generated by a finite set $S$, then the sets of the form

$$ \mathcal{N}([\alpha, v], R) := \{[\alpha', v'] : (\mathbf{Sch}(\alpha, S), v) \cong_R (\mathbf{Sch}(\alpha', S), v')\}, \quad (2.7) $$

constitute a basis of clopen neighborhoods of $[\alpha, v]$.

## 2.3 Bass-Serre theory

Associated with the standard HNN-presentation of

$$ \mathrm{BS}(m, n) = \langle b, t | tb^m t^{-1} = b^n \rangle, $$

we have the $\mathrm{BS}(m, n)$-action on its Bass-Serre tree $\mathcal{T}$. Recall that $\mathcal{T}$ is the oriented tree with $V(\mathcal{T}) = \mathrm{BS}(m, n)/\langle b \rangle$, $E^+(\mathcal{T}) = \mathrm{BS}(m, n)/\langle b^n \rangle$,

$$ \mathfrak{s}(\gamma \langle b^n \rangle) = \gamma \langle b \rangle, \quad \text{and} \quad \mathfrak{t}(\gamma \langle b^n \rangle) = \gamma t \langle b \rangle $$

and given a subgroup $\Lambda \leq \mathrm{BS}(m, n)$, the quotient $\Lambda \backslash \mathcal{T}$ has the structure of a graph of groups whose fundamental group is $\Lambda$, see [Ser80].

**Remark 2.8.** Let $\Lambda \leq \mathrm{BS}(m, n)$ be a subgroup. If $\Lambda \cap \langle b \rangle = \{\mathrm{id}\}$, then $\Lambda$ acts freely on $\mathcal{T}$; thus it is the fundamental group of the quotient graph $\Lambda \backslash \mathcal{T}$, hence $\Lambda$ is a free group.

Let us now concentrate on a subgroup $\Lambda \leq \mathrm{BS}(m, n)$ such that $\Lambda \cap \langle b \rangle \neq \{\mathrm{id}\}$. Then for the induced action $\Lambda \to \mathcal{T}$, each edge and vertex stabilizer is infinite cyclic: the tree $\mathcal{T}$ is a GBS-tree (for Generalized Baumslag-Solitar), in the sense of [For06, Lev07]. One can use this point of view to understand the graph of groups description of $\Lambda$. However, taking advantage of the transitivity of the $\mathrm{BS}(m, n)$-action on the edges and the vertices, we provide a slightly more precise description.

**Proposition 2.9.** Let $m$ and $n$ be non-zero integers. Let $\Lambda \leq \mathrm{BS}(m, n)$ be a subgroup such that $\Lambda \cap \langle b \rangle \neq \{\mathrm{id}\}$. The quotient graph of groups arising from the action $\Lambda \to \mathcal{T}$ is isomorphic to the graph of groups obtained by attaching a copy of $\mathbb{Z}$ to every vertex and every edge of the quotient graph $\Lambda \backslash \mathcal{T}$, with structural maps of positive edges

$$ \mathbb{Z}_e \hookrightarrow \mathbb{Z}_{\mathfrak{s}(e)}, \quad k \mapsto \frac{n}{\mathrm{deg}_{\mathrm{out}}(\mathfrak{s}(e))} \cdot k, $$

$$ \mathbb{Z}_e \hookrightarrow \mathbb{Z}_{\mathfrak{t}(e)}, \quad k \mapsto \frac{m}{\mathrm{deg}_{\mathrm{in}}(\mathfrak{t}(e))} \cdot k. $$
where $g \in \Lambda$ is any element such that $g \cdot t(\tilde{e}) = \tilde{t}(e)$ and the map $G_e \hookrightarrow G_{s(e)}$ is similar. This is unambiguous by (2.10).

Let us call orientation of an infinite cyclic group the choice of one generator (over two). This provides an identification to $\mathbb{Z}$. Once every stabilizer is oriented, the inclusions $G_e \hookrightarrow G_{s(e)}$ and $G_e \hookrightarrow G_{t(e)}$ become multiplications by non-zero integers $\lambda_\Lambda^-(e)$ and $\lambda_\Lambda^+(e)$, respectively. It now suffices to prove that, for well-chosen orientations, one has

$$ \lambda_\Lambda^-(e) = \frac{n}{\deg_{\text{out}}(\mathbf{s}(e))} \quad \text{and} \quad \lambda_\Lambda^+(e) = \frac{m}{\deg_{\text{in}}(\mathbf{t}(e))} \qquad (2.12) $$

for every positive edge $e \in E^+(\Lambda \setminus \mathcal{T})$.

Let us first observe that the absolute value of $\lambda_\Lambda^\pm(e)$ does not depend on the orientations: it is equal to $[G_v : G_e]$. In other words, if $\tilde{e}$ is a lift of $e$, $\tilde{v} := \mathbf{s}(\tilde{e}) = \text{and} \tilde{w} := \mathbf{t}(\tilde{e})$, we have

$$ |\lambda_\Lambda^-(e)| = [\text{Stab}_\Lambda(\tilde{v}) : \text{Stab}_\Lambda(\tilde{e})] = |\text{Stab}_\Lambda(\tilde{v}) \cdot \tilde{e}| \qquad (2.13) $$

$$ |\lambda_\Lambda^+(e)| = [\text{Stab}_\Lambda(\tilde{w}) : \text{Stab}_\Lambda(\tilde{e})] = |\text{Stab}_\Lambda(\tilde{w}) \cdot \tilde{e}|. \qquad (2.14) $$

Let $E_{\text{out}}(\tilde{v})$ be the set of $\tilde{v}$-outgoing edges. Its cardinal is $|E_{\text{out}}(\tilde{v})| = |n|$. Any generator of $\text{Stab}_\Gamma(\tilde{v})$ acts as a single $|n|$-cycle on $E_{\text{out}}(\tilde{v})$. Hence $E_{\text{out}}(\tilde{v})$ splits into $\text{Stab}_\Lambda(\tilde{v})$-orbits of equal size, that is $|\lambda_\Lambda^-(e)|$ according to (2.13). The number of these $\text{Stab}_\Lambda(\tilde{v})$-orbits is $\deg_{\text{out}}(v)$, thus $|n| = |\lambda_\Lambda^-(e)| \cdot \deg_{\text{out}}(v)$. We obtain similarly $|m| = |\lambda_\Lambda^+(e)| \cdot \deg_{\text{in}}(w)$, using ingoing edges and (2.14). We have established that (2.12) holds in absolute value.

Let us now turn to the signs in (2.12), for which we need explicit orientations of the $\Lambda$-stabilizers. We actually start by orienting the $\Gamma$-stabilizers.

Pick the vertex $\tilde{u}_0 := \langle b \rangle \in V(\mathcal{T})$, then $\text{Stab}_\Gamma(\tilde{u}_0) = \langle b \rangle$ and the edge $\tilde{d}_0 := \langle b^n \rangle \in E^+(\mathcal{T})$ has source $\tilde{u}_0$ and target $t\tilde{u}_0$. Since the $\Gamma$-stabilizers are canonically pairwise identified by conjugation (2.10), these choices induce a canonical conjugation-invariant orientation $x_*$ of all the vertex and edge $\Gamma$-stabilizers: $x_{g\tilde{u}_0} := gbg^{-1}$ for $\text{Stab}_\Gamma(g\tilde{u}_0)$ and $x_{g\tilde{d}_0} := gb^n g^{-1}$ for $\text{Stab}_\Gamma(g\tilde{d}_0)$.

The inclusions $\text{Stab}_\Gamma(\tilde{e}) \hookrightarrow \text{Stab}_\Gamma(\mathbf{s}(\tilde{e}))$ and $\text{Stab}_\Gamma(\tilde{e}) \hookrightarrow \text{Stab}_\Gamma(\mathbf{t}(\tilde{e}))$ become multiplications by non-zero integers that we denote by $\mu_\Gamma^-(\tilde{e})$ and $\mu_\Gamma^+(\tilde{e})$. We have $\mu_\Gamma^-(\tilde{e}) = n$ since $x_{\tilde{e}} = x_{\mathbf{s}(\tilde{e})}^n$ and $\mu_\Gamma^+(\tilde{e}) = m$ since

$$ x_{\tilde{e}} = gb^n g^{-1} = g(tbt^{-1})^m g^{-1} = x_{\mathbf{t}(\tilde{e})}^m. $$

The $\Lambda$-stabilizers have finite index in the corresponding $\Gamma$-stabilizers. We orient them coherently with the ambient $\Gamma$-stabilizers by using positive powers. The $\Lambda$-conjugations between $\Lambda$-stabilizers remain orientation-preserving,
Proof. In this proof we set $\Gamma := \mathrm{BS}(m, n)$. Let us consider the action of $\Lambda$ on the tree $\mathcal{T}$. Since $\mathcal{T}$ is locally finite, any edge adjacent to a vertex with infinite stabilizer has itself infinite stabilizer. It follows that all vertex and edge $\Lambda$-stabilizers are infinite. Being subgroups of the $\Gamma$-stabilizers, they are all isomorphic to $\mathbb{Z}$.

Observe that since $\Gamma$ acts transitively and the $\Gamma$-stabilizers are abelian, the $\Gamma$-stabilizers are canonically pairwise isomorphic: given any vertex $u \in V(\mathcal{T})$ and $a \in \mathrm{Stab}_{\Gamma}(u)$, one has

$$gag^{-1} = hah^{-1} \quad \text{for any } g, h \in \Gamma \text{ such that } gu = hu. \qquad (2.10)$$

Indeed since $h^{-1}g \in \mathrm{Stab}_{\Gamma}(u)$, we get that $h^{-1}gag^{-1}h = a$.

We now focus on the quotient graph of groups arising from the action $\Lambda \rightsquigarrow \mathcal{T}$. Let us recall from [Ser80] that its vertex groups are $G_v := \mathrm{Stab}_{\Lambda}(\tilde{v})$ and edge groups are $G_e := \mathrm{Stab}_{\Lambda}(\tilde{e})$, where $\tilde{v}, \tilde{e}$ are some lifts of $v, e$ in $\mathcal{T}$. Given any $e \in E^{+}(\Lambda \setminus \mathcal{T})$, the structural map $G_e \hookrightarrow G_{\mathfrak{t}(e)}$ is

$$G_e = \mathrm{Stab}_{\Lambda}(\tilde{e}) \hookrightarrow \mathrm{Stab}_{\Lambda}(\mathfrak{t}(\tilde{e})) \to \mathrm{Stab}_{\Lambda}(\widetilde{\mathfrak{t}(e)}) = G_{\mathfrak{t}(e)} \qquad (2.11) \\ a \mapsto a \mapsto gag^{-1}$$

where $g \in \Lambda$ is any element such that $g \cdot \mathfrak{t}(\tilde{e}) = \widetilde{\mathfrak{t}(e)}$ and the map $G_e \hookrightarrow G_{\mathfrak{s}(e)}$ is similar. This is unambiguous by (2.10).

Let us call *orientation* of an infinite cyclic group the choice of one generator (over two). This provides an identification to $\mathbb{Z}$. Once every stabilizer is oriented, the inclusions $G_e \hookrightarrow G_{\mathfrak{s}(e)}$ and $G_e \hookrightarrow G_{\mathfrak{t}(e)}$ become multiplications by non-zero integers $\lambda_{\Lambda}^{\pm}(e)$ and $\lambda_{\Lambda}^{\pm}(e)$, respectively. It now suffices to prove that, for well-chosen orientations, one has

$$\lambda_{\Lambda}^{\pm}(e) = \frac{n}{\mathrm{deg}_{\mathrm{out}}(\mathfrak{s}(e))} \quad \text{and} \quad \lambda_{\Lambda}^{\pm}(e) = \frac{m}{\mathrm{deg}_{\mathrm{in}}(\mathfrak{t}(e))} \qquad (2.12)$$

for every positive edge $e \in E^{+}(\Lambda \setminus \mathcal{T})$.

Let us first observe that the absolute value of $\lambda_{\Lambda}^{\pm}(e)$ does not depend on the orientations: it is equal to $[G_v : G_e]$. In other words, if $\tilde{e}$ is a lift of $e$, $\tilde{v} := \mathfrak{s}(\tilde{e}) = \text{and } \tilde{w} := \mathfrak{t}(\tilde{e})$, we have

$$|\lambda_{\Lambda}^{\pm}(e)| = [\mathrm{Stab}_{\Lambda}(\tilde{v}) : \mathrm{Stab}_{\Lambda}(\tilde{e})] = |\mathrm{Stab}_{\Lambda}(\tilde{v}) \cdot \tilde{e}| \qquad (2.13)$$

$$|\lambda_{\Lambda}^{\pm}(e)| = [\mathrm{Stab}_{\Lambda}(\tilde{w}) : \mathrm{Stab}_{\Lambda}(\tilde{e})] = |\mathrm{Stab}_{\Lambda}(\tilde{w}) \cdot \tilde{e}|. \qquad (2.14)$$
therefore by (2.11) the inclusion $\text{Stab}_\Lambda(\tilde{e}) \hookrightarrow \text{Stab}_\Lambda(\mathfrak{t}(\tilde{e}))$ becomes the multiplication by $\lambda_\Lambda^+(e)$. Similarly, the inclusion $\text{Stab}_\Lambda(\tilde{e}) \hookrightarrow \text{Stab}_\Lambda(\mathfrak{s}(\tilde{e}))$ becomes multiplication by $\lambda_\Lambda^-(e)$. Since the orientations are coherent, we conclude that $\lambda_\Lambda^-(e)$ has the same sign as $\mu_\Gamma^-(e) = n$ and $\lambda_\Lambda^+(e)$ has the same sign as $\mu_\Gamma^+(e) = m$. $\square$

**Corollary 2.15.** Let $m$ and $n$ be non-zero integers. Let $\Lambda \leqslant \text{BS}(m, n)$ be a subgroup such that $\Lambda \cap \langle b \rangle \neq \{\text{id}\}$. The isomorphism type of $\Lambda$ is completely determined by the oriented graph $\Lambda \setminus \mathcal{T}$.

**Proposition 2.16.** Let $m$ and $n$ be non-zero integers and let $\Lambda \leqslant \text{BS}(m, n)$ be a subgroup.

1. If $\Lambda \cap \langle b \rangle \neq \{\text{id}\}$, then either $\Lambda \simeq \mathbb{Z}$ is virtually a subgroup of $\langle b \rangle$ or $\Lambda$ is not a free group.

2. If $|m| = 1$ or $|n| = 1$, then the fundamental group of the underlying graph $\Lambda \setminus \mathcal{T}$ has rank $\leq 1$.

The first item of the proposition follows from standard techniques in $\ell^2$-cohomology: if $\Lambda \cap \langle b \rangle \neq \{\text{id}\}$, then $\Lambda$ is the fundamental group of a graph of groups whose vertex and edge groups are isomorphic to $\mathbb{Z}$; all the $\ell^2$-Betti numbers of such a group vanish. For the comfort of the reader we propose a proof by hand.

*Proof.* We start with the first item. Recall that in a free group $F$, whenever non-trivial elements $g, h \in F$ satisfy $gh^kg^{-1} = h^l$ with $k \neq 0 \neq l$, then there is $a \in F$ such that $g, h$ are both powers of $a$. In particular, such elements $g, h$ always commute.

Now, assume that $\Lambda$ is free and $\Lambda \cap \langle b \rangle \neq \{\text{id}\}$, say $\Lambda \cap \langle b \rangle = \langle b^s \rangle$ where $s > 0$. Pick any $\lambda \in \Lambda$ and set $H_\lambda := \langle b^s \rangle \cap \lambda \langle b^s \rangle \lambda^{-1}$, which is the intersection of $\Lambda$ with the stabilizer of the geodesic $[\langle b \rangle, \lambda \langle b \rangle]$ in $\mathcal{T}$. Observe that $H_\lambda$ is a finite index subgroup of both $\langle b^s \rangle$ and $\lambda \langle b^s \rangle \lambda^{-1}$. Therefore, there are $k \neq 0 \neq l$ such that $\lambda b^{sk} \lambda^{-1} = b^{sl}$. As $\Lambda$ is free, $\lambda$ and $b^s$ commute.

Consequently, the center of $\Lambda$ contains $\langle b^s \rangle$. Thus, the rank of $\Lambda$ is 1; in other words $\Lambda$ is infinite cyclic. It is now clear that $\langle b^s \rangle$ has finite index in both $\Lambda$ and $\langle b \rangle$, so that $\Lambda$ is virtually a subgroup of $\langle b \rangle$.

Let us turn to the second item. The fundamental group of a graph of groups surjects onto the fundamental group of the underlying graph. The condition in Item 2 implies the amenability of $\text{BS}(m, n)$. It thus cannot surject onto a non-amenable free group. $\square$
Let $E_{\text{out}}(\tilde{v})$ be the set of $\tilde{v}$-outgoing edges. Its cardinal is $|E_{\text{out}}(\tilde{v})| = |n|$. Any generator of $\text{Stab}_{\Gamma}(\tilde{v})$ acts as a single $|n|$-cycle on $E_{\text{out}}(\tilde{v})$. Hence $E_{\text{out}}(\tilde{v})$ splits into $\text{Stab}_{\Lambda}(\tilde{v})$-orbits of equal size, that is $|\lambda_{\Lambda}^{-}(e)|$ according to (2.13). The number of these $\text{Stab}_{\Lambda}(\tilde{v})$-orbits is $\text{deg}_{\text{out}}(v)$, thus $|n| = |\lambda_{\Lambda}^{-}(e)| \cdot \text{deg}_{\text{out}}(v)$. We obtain similarly $|m| = |\lambda_{\Lambda}^{+}(e)| \cdot \text{deg}_{\text{in}}(w)$, using incoming edges and (2.14). We have established that (2.12) holds in absolute value.

Let us now turn to the signs in (2.12), for which we need explicit orientations of the $\Lambda$-stabilizers. We actually start by orienting the $\Gamma$-stabilizers.

Pick the vertex $\tilde{u}_0 := \langle b \rangle \in V(\mathcal{T})$, then $\text{Stab}_{\Gamma}(\tilde{u}_0) = \langle b \rangle$ and the edge $\tilde{d}_0 := \langle b^n \rangle \in E^{+}(\mathcal{T})$ has source $\tilde{u}_0$ and target $t\tilde{u}_0$. Since the $\Gamma$-stabilizers are canonically pairwise identified by conjugation (2.10), these choices induce a canonical conjugation-invariant orientation $x_*$ of all the vertex and edge $\Gamma$-stabilizers: $x_{g\tilde{u}_0} := gbg^{-1}$ for $\text{Stab}_{\Gamma}(g\tilde{u}_0)$ and $x_{g\tilde{d}_0} := gb^n g^{-1}$ for $\text{Stab}_{\Gamma}(g\tilde{d}_0)$.

The inclusions $\text{Stab}_{\Gamma}(\tilde{e}) \hookrightarrow \text{Stab}_{\Gamma}(\mathfrak{s}(\tilde{e}))$ and $\text{Stab}_{\Gamma}(\tilde{e}) \hookrightarrow \text{Stab}_{\Gamma}(\mathfrak{t}(\tilde{e}))$ become multiplications by non-zero integers that we denote by $\mu_{\Gamma}^{-}(\tilde{e})$ and $\mu_{\Gamma}^{+}(\tilde{e})$. We have $\mu_{\Gamma}^{-}(\tilde{e}) = n$ since $x_{\tilde{e}} = x_{\mathfrak{s}(\tilde{e})}^n$ and $\mu_{\Gamma}^{+}(\tilde{e}) = m$ since

$$x_{\tilde{e}} = gb^n g^{-1} = g(tbt^{-1})^m g^{-1} = x_{\mathfrak{t}(\tilde{e})}^m.$$

The $\Lambda$-stabilizers have finite index in the corresponding $\Gamma$-stabilizers. We orient them coherently with the ambient $\Gamma$-stabilizers by using positive powers. The $\Lambda$-conjugations between $\Lambda$-stabilizers remain orientation-preserving, therefore by (2.11) the inclusion $\text{Stab}_{\Lambda}(\tilde{e}) \hookrightarrow \text{Stab}_{\Lambda}(\mathfrak{t}(\tilde{e}))$ becomes the multiplication by $\lambda_{\Lambda}^{+}(e)$. Similarly, the inclusion $\text{Stab}_{\Lambda}(\tilde{e}) \hookrightarrow \text{Stab}_{\Lambda}(\mathfrak{s}(\tilde{e}))$ becomes multiplication by $\lambda_{\Lambda}^{-}(e)$. Since the orientations are coherent, we conclude that $\lambda_{\Lambda}^{-}(e)$ has the same sign as $\mu_{\Gamma}^{-}(e) = n$ and $\lambda_{\Lambda}^{+}(e)$ has the same sign as $\mu_{\Gamma}^{+}(e) = m$. $\square$

**Corollary 2.15.** Let $m$ and $n$ be non-zero integers. Let $\Lambda \leq \text{BS}(m, n)$ be a subgroup such that $\Lambda \cap \langle b \rangle \neq \{\text{id}\}$. The isomorphism type of $\Lambda$ is completely determined by the oriented graph $\Lambda \setminus \mathcal{T}$. $\square$

**Proposition 2.16.** Let $m$ and $n$ be non-zero integers and let $\Lambda \leq \text{BS}(m, n)$ be a subgroup.

1. If $\Lambda \cap \langle b \rangle \neq \{\text{id}\}$, then either $\Lambda \simeq \mathbb{Z}$ is virtually a subgroup of $\langle b \rangle$ or $\Lambda$ is not a free group.

2. If $|m| = 1$ or $|n| = 1$, then the fundamental group of the underlying graph $\Lambda \setminus \mathcal{T}$ is a free group of rank $\leq 1$.
# 3 Bass-Serre graphs

## 3.1 Pre-actions

Let $m, n \in \mathbb{Z}\{0\}$. Recall that $\text{BS}(m, n) = \langle b, t \mid tb^m = b^nt \rangle$ and that our actions are on the right. Accordingly, in a product of (partial) bijections $\sigma, \tau$, $\sigma$ is applied first.

**Definition 3.1.** Given a bijection $\beta$ of a set $X$ and a partial bijection $\tau$ of $X$, we say that $\tau$ is $(\beta^n, \beta^m)$-equivariant if $\tau\beta^m = \beta^n\tau$ as partial bijections, that is:

* $\text{dom}(\tau)$ is $\beta^n$-invariant;

* $\text{rang}(\tau)$ is $\beta^m$-invariant;

* $x\tau\beta^m = x\beta^n\tau$ for all $x \in \text{dom}(\tau)$.

A pre-action of $\text{BS}(m, n)$ on a set $X$ is a couple $(\beta, \tau)$ where $\beta$ is a bijection of $X$ and $\tau$ is a $(\beta^n, \beta^m)$-equivariant partial bijection of $X$. The set $X$ is called the **domain** of the pre-action. Such a pre-action is **saturated** if $\text{dom}(\tau) = X = \text{rang}(\tau)$.

**Remark 3.2.** Saturated pre-actions $(\beta, \tau)$ correspond to actions $\alpha$ of $\text{BS}(m, n)$ on the same set $X$ under the association $\beta \leftrightarrow \alpha(b)$ and $\tau \leftrightarrow \alpha(t)$.

**Definition 3.3.** Given a pre-action $(\beta, \tau)$ of $\text{BS}(m, n)$, its **Schreier graph** is the oriented labeled graph $\text{Sch}(\beta, \tau) = \mathcal{G}$ defined by

$$V(\mathcal{G}) := X, \quad \left\{ \begin{array}{l} E^{+}(\mathcal{G}) := X \times \{b\} \sqcup \text{dom}(\tau) \times \{t\}, \\ E^{-}(\mathcal{G}) := X \times \{b^{-1}\} \sqcup \text{rang}(\tau) \times \{t^{-1}\}, \end{array} \right.$$

where the label of any edge is its second component and:

* for all $x \in X$, we set

$$\mathfrak{s}(x, b) := x, \quad \mathfrak{t}(x, b) := x\beta, \quad \text{and} \quad \overline{(x, b)} := (x\beta, b^{-1});$$

* for all $x \in \text{dom}(\tau)$, we set

$$\mathfrak{s}(x, t) := x, \quad \mathfrak{t}(x, t) := x\tau, \quad \text{and} \quad \overline{(x, t)} := (x\tau, t^{-1}).$$

Notice that the orientation of any edge $(x, l)$ is determined by its label $l$ and that the source of $(x, l)$ is $x$, regardless of its orientation.

Noting that a $\text{BS}(m, n)$-action is transitive if and only if the associated Schreier graph is connected, we make the following definition.

**Definition 3.4.** A pre-action of $\text{BS}(m, n)$ is **transitive** if its Schreier graph is connected.
If $\Lambda \cap \langle b \rangle = \{\mathrm{id}\}$, then $\Lambda$ is the fundamental group of the underlying graph $\Lambda \setminus \mathcal{T}$ (see Remark 2.8).

The first item of the proposition follows from standard techniques in $\ell^2$-cohomology: if $\Lambda \cap \langle b \rangle \neq \{\mathrm{id}\}$, then $\Lambda$ is the fundamental group of a graph of groups whose vertex and edge groups are isomorphic to $\mathbb{Z}$; all the $\ell^2$-Betti numbers of such a group vanish. For the comfort of the reader we propose a proof by hand.

*Proof.* We start with the first item. Recall that in a free group $F$, whenever non-trivial elements $g, h \in F$ satisfy $gh^k g^{-1} = h^l$ with $k \neq 0 \neq l$, then there is $a \in F$ such that $g, h$ are both powers of $a$. In particular, such elements $g, h$ always commute.

Now, assume that $\Lambda$ is free and $\Lambda \cap \langle b \rangle \neq \{\mathrm{id}\}$, say $\Lambda \cap \langle b \rangle = \langle b^s \rangle$ where $s > 0$. Pick any $\lambda \in \Lambda$ and set $H_\lambda := \langle b^s \rangle \cap \lambda \langle b^s \rangle \lambda^{-1}$, which is the intersection of $\Lambda$ with the stabilizer of the geodesic $[\langle b \rangle, \lambda \langle b \rangle]$ in $\mathcal{T}$. Observe that $H_\lambda$ is a finite index subgroup of both $\langle b^s \rangle$ and $\lambda \langle b^s \rangle \lambda^{-1}$. Therefore, there are $k \neq 0 \neq l$ such that $\lambda b^{sk} \lambda^{-1} = b^{sl}$. As $\Lambda$ is free, $\lambda$ and $b^s$ commute.

Consequently, the center of $\Lambda$ contains $\langle b^s \rangle$. Thus, the rank of $\Lambda$ is 1; in other words $\Lambda$ is infinite cyclic. It is now clear that $\langle b^s \rangle$ has finite index in both $\Lambda$ and $\langle b \rangle$, so $\Lambda$ is virtually a subgroup of $\langle b \rangle$.

Let us turn to the second item. The fundamental group of a graph of groups surjects onto the fundamental group of the underlying graph. The condition in Item 2 implies the amenability of $\mathrm{BS}(m, n)$. Its subgroups thus cannot surject onto a non-amenable free group. $\square$

# 3 Bass-Serre graphs

## 3.1 Pre-actions

Let $m, n \in \mathbb{Z} \setminus \{0\}$ and $\mathrm{BS}(m, n) = \langle b, t \mid tb^m = b^n t \rangle$.

Recall that a partial bijection of a set $X$ is a bijection between two subsets of $X$. Our actions are on the right; thus in a product of (partial) bijections $\sigma\tau$, the transformation $\sigma$ is applied first.

**Definition 3.1.** Given a bijection $\beta$ of a set $X$ and a partial bijection $\tau$ of $X$, we say that $\tau$ is $(\beta^n, \beta^m)$-equivariant if $\tau\beta^m = \beta^n\tau$ as partial bijections, that is:

* $\mathrm{dom}(\tau)$ is $\beta^n$-invariant;
## 3.2 Bass-Serre graphs

We now introduce an important tool for our study. It is the labeled graph obtained from the Schreier graph defined in Section 3.1 by “shrinking each $\beta$-orbit to one point”. We identify together the $t$-edges whose initial vertices belong to the same $\beta^n$-orbit. Note that their terminal vertices automatically belong to the same $\beta^m$-orbit.

We label the vertices by the cardinality of the corresponding $\beta$-orbit and the edges by the cardinality of the corresponding $\beta^n$-orbit. This is illustrated by Figure 2. The formal definition is as follows.

**Definition 3.5.** The Bass-Serre graph associated to a pre-action $\alpha = (\beta, \tau)$ of $\text{BS}(m, n)$ on a set $X$ is the oriented labeled graph $\mathbf{BS}(\alpha)$ defined by

$$V(\mathbf{BS}(\alpha)) := X/\langle\beta\rangle, \quad \left\{ \begin{array}{l} E^{+}(\mathbf{BS}(\alpha)) := \text{dom}(\tau)/\langle\beta^n\rangle, \\ E^{-}(\mathbf{BS}(\alpha)) := \text{rang}(\tau)/\langle\beta^m\rangle. \end{array} \right.$$

For every $x \in \text{dom}\,\tau$, we set

$$\mathbf{s}(x \langle\beta^n\rangle) := x \langle\beta\rangle, \quad \mathbf{t}(x \langle\beta^n\rangle) := x\tau \langle\beta\rangle, \quad \text{and} \quad \overline{x \langle\beta^n\rangle} := x\tau \langle\beta^m\rangle = x \langle\beta^n\rangle \tau.$$

We define the label map $L: V(\mathbf{BS}(\alpha)) \sqcup E(\mathbf{BS}(\alpha)) \to \mathbb{Z}_{\ge 1} \cup \{\infty\}$ by

$$L(x \langle\beta\rangle) := |x \langle\beta\rangle|, \quad L(x \langle\beta^n\rangle) := |x \langle\beta^n\rangle|, \quad L(y \langle\beta^m\rangle) := |y \langle\beta^m\rangle|.$$

**Remark 3.6.** For any $x \in \text{dom}(\tau)$, the $(\beta^n, \beta^m)$-equivariant partial bijection $\tau$ induces a bijection from $x\langle\beta^n\rangle$ to $x\tau\langle\beta^m\rangle$. Thus both the target and the opposite maps of $\mathbf{BS}(\alpha)$ are well-defined and the label of each edge is equal to the label of its opposite.

**Remark 3.7.** We view the sets $E^+(\mathbf{BS}(\alpha))$ and $E^-(\mathbf{BS}(\alpha))$ as disjoint sets, even though we might have that $\text{dom}(\tau)/\langle\beta^n\rangle \cap \text{rang}(\tau)/\langle\beta^m\rangle \neq \emptyset$. Note that the source of an edge $x\langle\beta^k\rangle \in E^+(\mathbf{BS}(\alpha))$ is $x\langle\beta\rangle$ regardless of its orientation.

**Remark 3.8.** The groups $\text{BS}(m, n)$ and $\text{BS}(n, m)$ are isomorphic via $b \mapsto b$ and $t \mapsto t^{-1}$. For every pre-action $(\beta, \tau)$ of $\text{BS}(m, n)$, the couple $(\beta, \tau^{-1})$ is a pre-action of $\text{BS}(n, m)$. At the level of Bass-Serre graphs, $\mathbf{BS}(\beta, \tau)$ and $\mathbf{BS}(\beta, \tau^{-1})$ coincide, except that the orientation is reversed.

**Remark 3.9.** In the case of a transitive $\text{BS}(m, n)$-action, the graph underlying our Bass-Serre graph is the quotient of the Bass-Serre tree $\mathcal{T}$ by the stabilizer of any point of $X$, as will be explained Section 3.6.
• `rng(τ)` is $β^m$-invariant;

• $x\tau\beta^m = x\beta^n\tau$ for all $x \in \text{dom}(\tau)$.

A pre-action of $\text{BS}(m, n)$ on a set $X$ is a couple $(\beta, \tau)$ where $\beta$ is a bijection of $X$ and $\tau$ is a $(\beta^n, \beta^m)$-equivariant partial bijection of $X$. The set $X$ is called the **domain** of the pre-action. Such a pre-action is **saturated** if $\text{dom}(\tau) = X = \text{rang}(\tau)$.

**Remark 3.2.** Saturated pre-actions $(\beta, \tau)$ correspond to actions $\alpha$ of $\text{BS}(m, n)$ on the same set $X$ under the association $\beta \leftrightarrow \alpha(b)$ and $\tau \leftrightarrow \alpha(t)$.

**Definition 3.3.** Given a pre-action $(\beta, \tau)$ of $\text{BS}(m, n)$, its **Schreier graph** is the oriented labeled graph $\text{Sch}(\beta, \tau) = \mathcal{G}$ defined by

$$V(\mathcal{G}) := X, \quad \left\{ \begin{array}{l} E^{+}(\mathcal{G}) := X \times \{b\} \sqcup \text{dom}(\tau) \times \{t\}, \\ E^{-}(\mathcal{G}) := X \times \{b^{-1}\} \sqcup \text{rang}(\tau) \times \{t^{-1}\}, \end{array} \right.$$

where the label of any edge is its second component and:

• for all $x \in X$, we set

$$\mathfrak{s}(x, b) := x, \quad \mathfrak{t}(x, b) := x\beta, \quad \text{and} \quad \overline{(x, b)} := (x\beta, b^{-1});$$

• for all $x \in \text{dom}(\tau)$, we set

$$\mathfrak{s}(x, t) := x, \quad \mathfrak{t}(x, t) := x\tau, \quad \text{and} \quad \overline{(x, t)} := (x\tau, t^{-1}).$$

Notice that the orientation of any edge $(x, l)$ is determined by its label $l$ and that the source of $(x, l)$ is $x$, regardless of its orientation.

Noting that a $\text{BS}(m, n)$-action is transitive if and only if the associated Schreier graph is connected, we make the following definition.

**Definition 3.4.** A pre-action of $\text{BS}(m, n)$ is **transitive** if its Schreier graph is connected.

## 3.2 Bass-Serre graphs

We now introduce an important tool for our study. It is the labeled graph obtained from the Schreier graph defined in Section 3.1 by “shrinking each $\beta$-orbit to one point”. We identify together the $t$-edges whose initial vertices belong to the same $\beta^n$-orbit. Note that their terminal vertices automatically belong to the same $\beta^m$-orbit.

We label the vertices by the cardinality of the corresponding $\beta$-orbit and the edges by the cardinality of the corresponding $\beta^n$-orbit. This is illustrated by Figure 2. The formal definition is as follows.
We now clarify what we meant by “shrinking each $\beta$-orbit to a point”, by noting that we have the following simplicial map from the Schreier graph to the Bass-Serre graph of any pre-action.

**Definition 3.10.** The projection associated to a pre-action $\alpha = (\beta, \tau)$ is the application $\pi_{\alpha}$ given by

$$ 
\begin{align*}
V(\mathbf{Sch}(\alpha)) &\to V(\mathbf{BS}(\alpha)), & x &\mapsto x \langle \beta \rangle \\
E_t^+(\mathbf{Sch}(\alpha)) &\to E^+(\mathbf{BS}(\alpha)), & (x, t) &\mapsto x \langle \beta^n \rangle \\
E_t^-(\mathbf{Sch}(\alpha)) &\to E^-(\mathbf{BS}(\alpha)), & (x, t^{-1}) &\mapsto x \langle \beta^m \rangle \\
E_b(\mathbf{Sch}(\alpha)) &\to V(\mathbf{BS}(\alpha)), & (x, b^{\pm 1}) &\mapsto x \langle \beta \rangle
\end{align*}
$$

where $E_t^{\pm}(\mathbf{Sch}(\alpha))$ is the subset of edges in $\mathbf{Sch}(\alpha)$ whose label is $t$ or $t^{-1}$ respectively and $E_b$ is the subset of edges whose label is $b$ or $b^{-1}$.

This projection is illustrated in Figure 2. Given any subgraph $\mathcal{G} \subseteq \mathbf{Sch}(\alpha)$ or path $\mathfrak{p}$ in $\mathbf{Sch}(\alpha)$ we obtain a subgraph $\pi_{\alpha}(\mathcal{G}) \subseteq \mathbf{BS}(\alpha)$ or a path $\pi_{\alpha}(\mathfrak{p})$ in $\mathbf{BS}(\alpha)$.

<figure><img src="image_3.png" /><figcaption>Figure 2: The projection from the Schreier graph onto the Bass-Serre graph of some non-saturated transitive $\mathbf{BS}(2, 3)$-pre-action. The dotted circles represent the $\beta$-orbits in the Schreier graph.</figcaption></figure>

Note that for every vertex $v = x \langle \beta \rangle$,

$$ |x \langle \beta^k \rangle| = \frac{|x \langle \beta \rangle|}{\gcd(|x \langle \beta \rangle|, k)}, $$
**Definition 3.5.** The **Bass-Serre graph** associated to a pre-action $\alpha = (\beta, \tau)$ of $\text{BS}(m, n)$ on a set $X$ is the oriented labeled graph $\text{BS}(\alpha)$ defined by

$$V(\mathbf{BS}(\alpha)) := X/\langle\beta\rangle, \quad \left\{ \begin{array}{l} E^{+}(\mathbf{BS}(\alpha)) := \text{dom}(\tau)/\langle\beta^n\rangle, \\ E^{-}(\mathbf{BS}(\alpha)) := \text{rang}(\tau)/\langle\beta^m\rangle. \end{array} \right.$$

For every $x \in \text{dom}\,\tau$, we set

$$\mathbf{s}(x \langle\beta^n\rangle) := x \langle\beta\rangle, \quad \mathbf{t}(x \langle\beta^n\rangle) := x\tau \langle\beta\rangle, \quad \text{and} \quad \overline{x \langle\beta^n\rangle} := x\tau \langle\beta^m\rangle = x \langle\beta^n\rangle\tau.$$

We define the label map $L: V(\mathbf{BS}(\alpha)) \sqcup E(\mathbf{BS}(\alpha)) \to \mathbb{Z}_{\ge 1} \cup \{\infty\}$ by

$$L(x \langle\beta\rangle) := |x \langle\beta\rangle|, \quad L(x \langle\beta^n\rangle) := |x \langle\beta^n\rangle|, \quad L(y \langle\beta^m\rangle) := |y \langle\beta^m\rangle|.$$

**Remark 3.6.** For any $x \in \text{dom}(\tau)$, the $(\beta^n, \beta^m)$-equivariant partial bijection $\tau$ induces a bijection from $x\langle\beta^n\rangle$ to $x\tau\langle\beta^m\rangle$. Thus both the target and the opposite maps of $\mathbf{BS}(\alpha)$ are well-defined and the label of each edge is equal to the label of its opposite.

**Remark 3.7.** We view the sets $E^+(\mathbf{BS}(\alpha))$ and $E^-(\mathbf{BS}(\alpha))$ as disjoint sets, even though we might have that $\text{dom}(\tau)/\langle\beta^n\rangle \cap \text{rang}(\tau)/\langle\beta^m\rangle \neq \emptyset$. Note that the source of an edge $x\langle\beta^k\rangle \in E^+(\mathbf{BS}(\alpha))$ is $x\langle\beta\rangle$ regardless of its orientation.

**Remark 3.8.** The groups $\text{BS}(m, n)$ and $\text{BS}(n, m)$ are isomorphic via $b \mapsto b$ and $t \mapsto t^{-1}$. For every pre-action $(\beta, \tau)$ of $\text{BS}(m, n)$, the couple $(\beta, \tau^{-1})$ is a pre-action of $\text{BS}(n, m)$. At the level of Bass-Serre graphs, $\mathbf{BS}(\beta, \tau)$ and $\mathbf{BS}(\beta, \tau^{-1})$ coincide, except that the orientation is reversed.

**Remark 3.9.** In the case of a transitive $\text{BS}(m, n)$-action, the graph underlying our Bass-Serre graph is the quotient of the Bass-Serre tree $\mathcal{T}$ by the stabilizer of any point of $X$, as will be explained in Section 3.6.

We now clarify what we meant by “shrinking each $\beta$-orbit to a point”, by noting that we have the following simplicial map from the Schreier graph to the Bass-Serre graph of any pre-action.

**Definition 3.10.** The **projection** associated to a pre-action $\alpha = (\beta, \tau)$ is the application $\pi_\alpha$ given by

$$\begin{align*}
V(\mathbf{Sch}(\alpha)) &\to V(\mathbf{BS}(\alpha)), & x &\mapsto x \langle\beta\rangle \\
E_t^+(\mathbf{Sch}(\alpha)) &\to E^+(\mathbf{BS}(\alpha)), & (x, t) &\mapsto x \langle\beta^n\rangle \\
E_t^-(\mathbf{Sch}(\alpha)) &\to E^-(\mathbf{BS}(\alpha)), & (x, t^{-1}) &\mapsto x \langle\beta^m\rangle \\
E_b(\mathbf{Sch}(\alpha)) &\to V(\mathbf{BS}(\alpha)), & (x, b^{\pm 1}) &\mapsto x \langle\beta\rangle
\end{align*}$$
thus the following facts hold:

* all the $v$-outgoing edges $e$ have the same label, which is:

$$L(e) = \frac{L(v)}{\gcd(L(v), n)},$$

* all the $v$-incoming edges $e'$ have the same label, which is:

$$L(e') = \frac{L(v)}{\gcd(L(v), m)}.$$

We also have the following relations between labels and degrees:

* The outgoing degree $\deg_{\text{out}}(v)$ is equal to the number of $\beta^n$-orbits contained in $x\langle\beta\rangle \cap \text{dom}(\tau)$. Recall that $\text{dom}(\tau)$ is $\beta^n$-invariant. Since $x\langle\beta\rangle$ contains exactly $\gcd(L(v), n)$ orbits under $\beta^n$, we get

$$\deg_{\text{out}}(v) \le \gcd(L(v), n),$$

with equality if and only if $x\langle\beta\rangle \subseteq \text{dom}(\tau)$.

* Similarly, the incoming degree $\deg_{\text{in}}(v)$ is equal to the number of $\beta^m$-orbits contained in $x\langle\beta\rangle \cap \text{rang}(\tau)$, so

$$\deg_{\text{in}}(v) \le \gcd(L(v), m),$$

with equality if and only if $x\langle\beta\rangle \subseteq \text{rang}(\tau)$.

**Remark 3.11.** As a consequence of the last two items, the pre-action is an action if and only if, for every vertex $v$,

$$\deg_{\text{out}}(v) = \gcd(L(v), n) \quad \text{and} \quad \deg_{\text{in}}(v) = \gcd(L(v), m).$$

### 3.3 $(m, n)$-graphs

We now introduce an axiomatization of the Bass-Serre graphs we obtain from pre-actions. Recall that by convention $\gcd(\infty, k) = |k|$ for all $k \ne 0$.

**Definition 3.12.** An $(m, n)$-graph is an oriented labeled graph $\mathcal{G} = (V, E)$ with label map $L: V \cup E \to \mathbb{Z}_{\ge 1} \cup \{\infty\}$ such that:

* for every positive edge $e \in E^+$, then

$$\frac{L(\mathfrak{s}(e))}{\gcd(L(\mathfrak{s}(e)), n)} = L(e) = \frac{L(\mathfrak{t}(e))}{\gcd(L(\mathfrak{t}(e)), m)}; \quad (3.13)$$
where $E_t^{\pm}(\mathbf{Sch}(\alpha))$ is the subset of edges in $\mathbf{Sch}(\alpha)$ whose label is $t$ or $t^{-1}$ respectively and $E_b$ is the subset of edges whose label is $b$ or $b^{-1}$.

This projection is illustrated in Figure 2. Given any subgraph $\mathcal{G} \subseteq \mathbf{Sch}(\alpha)$ or path $\mathbf{p}$ in $\mathbf{Sch}(\alpha)$ we obtain a subgraph $\pi_{\alpha}(\mathcal{G}) \subseteq \mathbf{BS}(\alpha)$ or a path $\pi_{\alpha}(\mathbf{p})$ in $\mathbf{BS}(\alpha)$.

<figure><img src="image_2.png" /><figcaption>Figure 2: The projection from the Schreier graph onto the Bass-Serre graph of some non-saturated transitive $\mathbf{BS}(2, 3)$-pre-action. The dotted circles represent the $\beta$-orbits in the Schreier graph.</figcaption></figure>

Note that for every vertex $v = x\langle\beta\rangle$,

$$|x\langle\beta^k\rangle| = \frac{|x\langle\beta\rangle|}{\gcd(|x\langle\beta\rangle|, k)},$$

thus the following facts hold:

* all the $v$-outgoing edges $e$ have the same label, which is:

$$L(e) = \frac{L(v)}{\gcd(L(v), n)},$$

* all the $v$-incoming edges $e'$ have the same label, which is:

$$L(e') = \frac{L(v)}{\gcd(L(v), m)}.$$
* for every negative edge $e \in E^-$, $L(e) = L(\bar{e})$;

* for every vertex $v \in V$, we have

$$ \deg_{\text{out}}(v) \le \gcd(L(v), n) \quad \text{and} \quad \deg_{\text{in}}(v) \le \gcd(L(v), m). \quad (3.14) $$

**Example 3.15.** The Bass-Serre graph of any pre-action of $BS(m, n)$ is an $(m, n)$-graph. The converse will be shown in Proposition 3.22.

**Remark 3.16.** Observe that an edge label is uniquely determined by the label of any of its vertices. The edge labels are thus redundant and are just calculation tools.

**Example 3.17.** Let us see how labels interact for $m = 2$ and $n = 3$. If $e$ is an edge in a $(2, 3)$-graph, then once we fix the label of one of the extremities, the other one can be chosen according to the Table 1, using Formula 3.13 for $L(e)$. The reader is invited to consult the webpage [WebTool] to see the kinds of local constraints which occur in general.

<table><caption>Table 1: How the label of the extremities impact each other</caption><tr><td>If $\gcd(L(s(e)), 2) = 1$<br>$L(t(e)) \in \{L(e), 2L(e)\}$</td><td>If $\gcd(L(s(e)), 2) = 2$<br>$L(t(e)) = 2L(e)$</td></tr><tr><td>If $\gcd(L(t(e)), 3) = 1$<br>$L(s(e)) \in \{L(e), 3L(e)\}$</td><td>If $\gcd(L(t(e)), 3) = 3$<br>$L(s(e)) = 3L(e)$</td></tr></table>

In Figure 3, we give an illustrative example.

**Remark 3.18.** As in Remark 3.8, every $(m, n)$-graph can be turned into an $(n, m)$-graph by flipping the orientations of its edges. Note that this operation does not affect the labels.

**Remark 3.19.** In a connected $(m, n)$-graph, the labels are, either all finite, or all $\infty$ by Equation (3.13). This will be made more precise in Proposition 4.6. Observe that any oriented graph $\mathcal{G}$ satisfying $\deg_{\text{in}}(v) \le m$ and $\deg_{\text{out}}(v) \le n$ for every $v \in V(\mathcal{G})$ becomes an $(m, n)$-graph if we set all the labels to be infinite. However one cannot always put finite labels, see Lemma 3.33.
We also have the following relations between labels and degrees:

* The outgoing degree $\deg_{\text{out}}(v)$ is equal to the number of $\beta^n$-orbits contained in $x\langle\beta\rangle \cap \text{dom}(\tau)$. Recall that $\text{dom}(\tau)$ is $\beta^n$-invariant. Since $x\langle\beta\rangle$ contains exactly $\text{gcd}(L(v), n)$ orbits under $\beta^n$, we get

$$ \deg_{\text{out}}(v) \le \text{gcd}(L(v), n), $$

with equality if and only if $x\langle\beta\rangle \subseteq \text{dom}(\tau)$.

* Similarly, the incoming degree $\deg_{\text{in}}(v)$ is equal to the number of $\beta^m$-orbits contained in $x\langle\beta\rangle \cap \text{rng}(\tau)$, so

$$ \deg_{\text{in}}(v) \le \text{gcd}(L(v), m), $$

with equality if and only if $x\langle\beta\rangle \subseteq \text{rng}(\tau)$.

**Remark 3.11.** As a consequence of the last two items, the pre-action is an action if and only if, for every vertex $v$,

$$ \deg_{\text{out}}(v) = \text{gcd}(L(v), n) \quad \text{and} \quad \deg_{\text{in}}(v) = \text{gcd}(L(v), m). $$

## 3.3 $(m, n)$-graphs

We now introduce an axiomatization of the Bass-Serre graphs we obtain from pre-actions. Recall that by convention $\text{gcd}(\infty, k) = |k|$ for all $k \ne 0$.

**Definition 3.12.** An $(m, n)$-graph is an oriented labeled graph $\mathcal{G} = (V, E)$ with label map $L: V \cup E \to \mathbb{Z}_{\ge 1} \cup \{\infty\}$ such that:

* for every positive edge $e \in E^+$,

$$ \frac{L(\mathbf{s}(e))}{\text{gcd}(L(\mathbf{s}(e)), n)} = L(e) = \frac{L(\mathbf{t}(e))}{\text{gcd}(L(\mathbf{t}(e)), m)}; \quad (3.13) $$

* for every negative edge $e \in E^-$, $L(e) = L(\bar{e})$;

* for every vertex $v \in V$, we have

$$ \deg_{\text{out}}(v) \le \text{gcd}(L(v), n) \quad \text{and} \quad \deg_{\text{in}}(v) \le \text{gcd}(L(v), m). \quad (3.14) $$

**Example 3.15.** The Bass-Serre graph of any pre-action of $\text{BS}(m, n)$ is an $(m, n)$-graph. The converse will be shown in Proposition 3.22.

**Remark 3.16.** Observe that an edge label is uniquely determined by the label of any of its vertices. The edge labels are thus redundant and are just calculation tools (see also Remark 4.7).
<figure><img src="image_4.png" /><figcaption>Figure 3: Two examples of (2, 3)-graphs.</figcaption></figure>

**Definition 3.20.** Let $\mathcal{G}$ be an $(m, n)$-graph. A vertex $v$ in $\mathcal{G}$ is **saturated** if the inequalities (3.14) are indeed equalities, i.e.

$$ \deg_{\text{out}}(v) = \gcd(L(v), n) \quad \text{and} \quad \deg_{\text{in}}(v) = \gcd(L(v), m). $$

The $(m, n)$-graph $\mathcal{G}$ is **saturated** if all its vertices are saturated.

**Example 3.21.** The Bass-Serre graph of a pre-action of $\text{BS}(m, n)$ is saturated if and only if the pre-action is an action.

## 3.4 Realizing $(m, n)$-graphs as Bass-Serre graphs

**Proposition 3.22.** Every $(m, n)$-graph $\mathcal{G}$ is the Bass-Serre graph of at least one pre-action of $\text{BS}(m, n)$. Any such pre-action is an action if and only if $\mathcal{G}$ is saturated.

The above proposition is a consequence of the following stronger statement where by definition, a **sub-(m, n)-graph** of an $(m, n)$-graph $\mathcal{G}$ is a subgraph $\mathcal{G}'$ labeled by the restriction of the label map of $\mathcal{G}$.

**Proposition 3.23 (Extension of pre-actions from $(m, n)$-graphs).** Let $\mathcal{G}_1$ be the Bass-Serre graph of a pre-action $\alpha_1$ and let $\mathcal{G}_2$ be an $(m, n)$-graph that contains $\mathcal{G}_1$ as a sub-(m, n)-graph. Then $\mathcal{G}_2$ is the Bass-Serre graph of a pre-action $\alpha_2$ that extends $\alpha_1$.

*Proof.* We start with a pre-action $(\beta_1, \tau_1)$ on $X_1$ which yields the Bass-Serre graph $\mathcal{G}_1$. Let $W := V(\mathcal{G}_2) \setminus V(\mathcal{G}_1)$ and $X_2 := X_1 \sqcup \bigsqcup_{v \in W} X_v$ where each $X_v$
**Example 3.17.** Let us see how labels interact for $m = 2$ and $n = 3$. If $e$ is an edge in a (2,3)-graph, then once we fix the label of one of the extremities, the other one can be chosen according to the Table 1, using Formula (3.13) for $L(e)$. The reader is invited to consult the webpage [CGLMS22] to see the kinds of local constraints which occur in general.

<table><caption>Table 1: How the label of the extremities impact each other</caption><tr><td>If gcd(L(s(e)), 2) = 1<br>L(t(e)) ∈ {L(e), 2L(e)}</td><td>If gcd(L(s(e)), 2) = 2<br>L(t(e)) = 2L(e)</td></tr><tr><td>If gcd(L(t(e)), 3) = 1<br>L(s(e)) ∈ {L(e), 3L(e)}</td><td>If gcd(L(t(e)), 3) = 3<br>L(s(e)) = 3L(e)</td></tr></table>

In Figure 3, we give an illustrative example.

<figure><img src="image_3.png" /><figcaption>Figure 3: Two examples of (2,3)-graphs.</figcaption></figure>

**Remark 3.18.** As in Remark 3.8, every $(m, n)$-graph can be turned into an $(n, m)$-graph by flipping the orientations of its edges. Note that this operation does not affect the labels.

**Remark 3.19.** In a connected $(m, n)$-graph, the labels are, either all finite, or all $\infty$ by Equation (3.13). This will be made more precise in Proposition 4.6. Observe that any oriented graph $\mathcal{G}$ satisfying $\deg_{\text{in}}(v) \le m$ and
is a set of cardinality $|X_v| = L(v)$. We extend $\beta_1$ to a permutation $\beta_2$ of $X_2$ by making it act as a cycle of length $L(v)$ on $X_v$.

By Zorn's lemma, it suffices to extend $\tau_1$ when $G_1$ only lacks one positive $G_2$-edge. So suppose $E^+(G_1) \cup \{e\} = E^+(G_2)$. Then by Inequation (3.14) from Definition 3.12,

$$ \deg_{\text{out}}^{G_1}(\mathbf{s}(e)) < \deg_{\text{out}}^{G_2}(\mathbf{s}(e)) \le \gcd(L(\mathbf{s}(e)), n) $$

and similarly

$$ \deg_{\text{in}}^{G_1}(\mathbf{t}(e)) < \deg_{\text{in}}^{G_2}(\mathbf{t}(e)) \le \gcd(L(\mathbf{t}(e)), m). $$

We can thus find a $\beta^n$-orbit $y \langle\beta^n\rangle$ contained in the $\beta$-orbit $\mathbf{s}(e) \langle\beta\rangle$ but disjoint from $\text{dom}(\tau_1)$ and a $\beta^m$-orbit $z \langle\beta^m\rangle$ contained in the $\beta$-orbit $\mathbf{t}(e) \langle\beta\rangle$ but disjoint from $\text{rang}(\tau_1)$.

Since these two orbits $y \langle\beta^n\rangle$ and $z \langle\beta^m\rangle$ share the same cardinal $L(e)$, we can define $\tau_2$ as an extension of $\tau_1$ which is also $(\beta^n, \beta^m)$-equivariant when restricted to $y \langle\beta^n\rangle$ by letting

$$ y\beta^{kn}\tau_2 = z\beta^{km} \text{ for all } k \in \mathbb{Z}. $$

By construction $\tau_2$ is the desired extension. $\square$

<figure><img src="image_5.png" /></figure>

The pre-action $\alpha_2$ arising in Proposition 3.23 is definitively not unique in general. In a forthcoming work, we will characterize which $(m, n)$-graphs arise as Bass-Serre graphs of continuum many non-isomorphic actions. In particular we will show that the $(m, n)$-graphs whose underlying graph have non-finitely generated fundamental group are of this kind. Such $(m, n)$-graphs always exist as soon as $|m| \ge 2$ and $|n| \ge 2$. Here we give a simple example of a graph associated to continuum many non-isomorphic actions for $n = m = 2$.

**Example 3.24.** Let $G$ be the (2, 2)-graph whose underlying graph is such that $V(G) = \mathbb{Z}$ and for every $z \in V(G)$ there are exactly two $z$-outgoing edges, one to $z$ and the other to $z+1$. That is, $G$ is a line where every vertex has an extra loop. We set the labels of $G$ to be all infinite.

Set $X := V(G) \times \mathbb{Z} \cong \mathbb{Z} \times \mathbb{Z}$. For every function $f: \mathbb{Z} \to \mathbb{Z}$ such that $\forall w < 0$, $f(w) = 0$ and $f(0) \ne 0$, we define an action $\alpha_f$ as follows: for all
$\deg_{\text{out}}(v) \le n$ for every $v \in V(\mathcal{G})$ becomes an $(m, n)$-graph if we set all the labels to be infinite. However one cannot always put finite labels, see Lemma 3.33.

**Definition 3.20.** Let $\mathcal{G}$ be an $(m, n)$-graph. A vertex $v$ in $\mathcal{G}$ is **saturated** if the inequalities (3.14) are indeed equalities, i.e.

$$ \deg_{\text{out}}(v) = \gcd(L(v), n) \quad \text{and} \quad \deg_{\text{in}}(v) = \gcd(L(v), m). $$

The $(m, n)$-graph $\mathcal{G}$ is **saturated** if all its vertices are saturated.

**Example 3.21.** The Bass-Serre graph of a pre-action of $\text{BS}(m, n)$ is saturated if and only if the pre-action is an action.

## 3.4 Realizing $(m, n)$-graphs as Bass-Serre graphs

**Proposition 3.22.** Every $(m, n)$-graph $\mathcal{G}$ is the Bass-Serre graph of at least one pre-action of $\text{BS}(m, n)$. Any such pre-action is an action if and only if $\mathcal{G}$ is saturated.

The above proposition is a consequence of the following stronger statement where by definition, a **sub-$(m, n)$-graph** of an $(m, n)$-graph $\mathcal{G}$ is a subgraph $\mathcal{G}'$ labeled by the restriction of the label map of $\mathcal{G}$.

**Proposition 3.23 (Extension of pre-actions from $(m, n)$-graphs).** Let $\mathcal{G}_1$ be the Bass-Serre graph of a pre-action $\alpha_1$ and let $\mathcal{G}_2$ be an $(m, n)$-graph that contains $\mathcal{G}_1$ as a sub-$(m, n)$-graph. Then $\mathcal{G}_2$ is the Bass-Serre graph of a pre-action $\alpha_2$ that extends $\alpha_1$.

*Proof.* We start with a pre-action $(\beta_1, \tau_1)$ on $X_1$ which yields the Bass-Serre graph $\mathcal{G}_1$. Let $W := V(\mathcal{G}_2) \setminus V(\mathcal{G}_1)$ and $X_2 := X_1 \sqcup \bigsqcup_{v \in W} X_v$ where each $X_v$ is a set of cardinality $|X_v| = L(v)$. We extend $\beta_1$ to a permutation $\beta_2$ of $X_2$ by making it act as a cycle of length $L(v)$ on $X_v$.

By Zorn's lemma, it suffices to extend $\tau_1$ when $\mathcal{G}_1$ only lacks one positive $\mathcal{G}_2$-edge. So suppose $E^+(\mathcal{G}_1) \sqcup \{e\} = E^+(\mathcal{G}_2)$. Then by Inequality (3.14) in Definition 3.12,

$$ \deg_{\text{out}}^{\mathcal{G}_1}(\mathbf{s}(e)) < \deg_{\text{out}}^{\mathcal{G}_2}(\mathbf{s}(e)) \le \gcd(L(\mathbf{s}(e)), n) $$

and similarly

$$ \deg_{\text{in}}^{\mathcal{G}_1}(\mathbf{t}(e)) < \deg_{\text{in}}^{\mathcal{G}_2}(\mathbf{t}(e)) \le \gcd(L(\mathbf{t}(e)), m). $$
<figure><img src="image_6.png" /><figcaption>Figure 4: The (2,2)-graph $\mathcal{G}$</figcaption></figure>

$$ (k, l) \in X $$

$$ (k, l)\alpha_f(b) := (k, l + 1); $$

$$ (k, l)\alpha_f(t) := \begin{cases} (k + 1, l) & \text{if } l \text{ is odd;} \\ (k, l + f(k)) & \text{if } l \text{ is even.} \end{cases} $$

It is easy to check that all $\alpha_f$ are actions of $\mathrm{BS}(2, 2)$ whose Bass-Serre graph is $\mathcal{G}$, that $\alpha_f$ and $\alpha_g$ are non-conjugate for $f \neq g$, and that there are continuum many such actions.

## 3.5 Additional properties of $(m, n)$-graphs

In this section, we collect some basic consequences of the definition of $(m, n)$-graphs. Observe that Equation (3.13) is equivalent to the fact that

$$ \max(|L(s(e))|_p - |n|_p, 0) = |L(e)|_p = \max(|L(t(e))|_p - |m|_p, 0) \quad (3.25) $$

from which we obtain the following.

**Remark 3.26.** Consider an oriented labeled graph $\mathcal{G} = (V, E)$ with label map $L: V \sqcup E \to \mathbb{Z}_{\ge 1}$ satisfying $L(\bar{e}) = L(e)$ for every edge $e$. The labeled graph $\mathcal{G}$ is an $(m, n)$-graph if and only if the following two conditions hold:

* for every positive edge $e$ and every prime $p$ such that $|L(e)|_p \ge 1$,

$$ |L(\mathbf{s}(e))|_p = |L(e)|_p + |n|_p \quad \text{and} \quad |L(\mathbf{t}(e))|_p = |L(e)|_p + |m|_p, \quad (3.27) $$

* for every positive edge $e$ and every prime $p$ such that $|L(e)|_p = 0$,

$$ 0 \le |L(\mathbf{s}(e))|_p \le |n|_p \quad \text{and} \quad 0 \le |L(\mathbf{t}(e))|_p \le |m|_p. \quad (3.28) $$

In particular, in an $(m, n)$-graph, $|L(\mathbf{s}(e))|_p > |n|_p$ if and only if $|L(\mathbf{t}(e))|_p > |m|_p$, and if one of these two equivalent conditions is met then

$$ |L(\mathbf{t}(e))|_p = |L(\mathbf{s}(e))|_p + |m|_p - |n|_p. \quad (3.29) $$
We can thus find a $\beta_2^n$-orbit $y\langle\beta_2^n\rangle$ contained in the $\beta_2$-orbit $s(e)$ but disjoint from $\text{dom}(\tau_1)$ and a $\beta_2^m$-orbit $z\langle\beta_2^m\rangle$ contained in the $\beta_2$-orbit $t(e)$ but disjoint from $\text{rang}(\tau_1)$.

Since these two orbits $y\langle\beta_2^n\rangle$ and $z\langle\beta_2^m\rangle$ share the same cardinal $L(e)$, we can define $\tau_2$ as an extension of $\tau_1$ which is also $(\beta_2^n, \beta_2^m)$-equivariant when restricted to $y\langle\beta_2^n\rangle$ by letting

$$y\beta_2^{kn}\tau_2 = z\beta_2^{km} \text{ for all } k \in \mathbb{Z}.$$

By construction $\tau_2$ is the desired extension. $\square$

The pre-action $\alpha_2$ arising in Proposition 3.23 is definitively not unique in general. In a forthcoming work, we will characterize which $(m, n)$-graphs arise as Bass-Serre graphs of continuum many non-isomorphic actions. In particular we will show that the $(m, n)$-graphs whose underlying graph has non-finitely generated fundamental group are of this kind. Such $(m, n)$-graphs always exist as soon as $|m| \ge 2$ and $|n| \ge 2$. Here we give a simple example of a graph associated to continuum many non-isomorphic actions for $n = m = 2$.

**Example 3.24.** Let $\mathcal{G}$ be the (2,2)-graph whose underlying graph is such that $V(\mathcal{G}) = \mathbb{Z}$ and for every $z \in V(\mathcal{G})$ there are exactly two $z$-outgoing edges, one to $z$ and the other to $z+1$. That is, $\mathcal{G}$ is a line where every vertex has an extra loop. We set the labels of $\mathcal{G}$ to be all infinite.

<figure><img src="image_4.png" /><figcaption>Figure 4: The (2,2)-graph $\mathcal{G}$</figcaption></figure>

Set $X := V(\mathcal{G}) \times \mathbb{Z} \cong \mathbb{Z} \times \mathbb{Z}$. For every function $f: \mathbb{Z} \to \mathbb{Z}$ such that $\forall w < 0, f(w) = 0$ and $f(0) \ne 0$, we define an action $\alpha_f$ as follows: for all $(k, l) \in X$

$$(k, l)\alpha_f(b) := (k, l + 1);$$

$$(k, l)\alpha_f(t) := \begin{cases} (k + 1, l) & \text{if } l \text{ is odd;} \\ (k, l + f(k)) & \text{if } l \text{ is even.} \end{cases}$$
**Lemma 3.30.** Fix a prime $p$ such that $|n|_p < |m|_p$ and let $\mathcal{G}$ be an $(m, n)$-graph. If $(e_k)_{k \ge 1}$ is any infinite path consisting of positive edges with $L(\mathbf{s}(e_1)) \ne \infty$ and $|L(\mathbf{s}(e_1))|_p > |n|_p$, then

$$ \lim_{k \to +\infty} |L(\mathbf{s}(e_k))|_p = +\infty. $$

If $(e_k)_{k \ge 1}$ is any infinite path consisting of negative edges with $L(\mathbf{s}(e_1)) \ne \infty$, then

$$ \limsup_{k \to +\infty} |L(\mathbf{s}(e_k))|_p < |m|_p. $$

*Proof.* If $(e_k)_{k \ge 1}$ is an infinite path consisting of positive edges such that $|L(\mathbf{s}(e_1))|_p > |n|_p$, then by a straightforward induction using Equation (3.29) we have that

$$ |L(\mathbf{s}(e_k))|_p = |L(\mathbf{s}(e_1))|_p + k(|m|_p - |n|_p) \quad (3.31) $$

for all $k \ge 1$. The first result follows.

For the second one, let $(e_k)_{k \ge 1}$ be an infinite path consisting of negative edges. By exchanging the roles in Equation (3.29), we have that if $|L(\mathbf{s}(e_k))|_p > |m|_p$ then $|L(\mathbf{s}(e_{k+1}))|_p < |L(\mathbf{s}(e_k))|_p$. So there must be $k_0 \in \mathbb{N}$ such that $|L(\mathbf{s}(e_{k_0}))|_p \le |m|_p$. We then have $|L(\mathbf{s}(e_{k_0+1}))|_p \le |n|_p < |m|_p$ so by induction $|L(\mathbf{s}(e_k))|_p < |m|_p$ for all $k \ge k_0$, which finishes the proof. $\square$

**Remark 3.32.** It follows from Equation (3.31) that any infinite path $(e_k)_{k \ge 1}$ consisting of positive edges with $L(\mathbf{s}(e_1)) \ne \infty$ and $|L(\mathbf{s}(e_1))|_p > |n|_p$ has to be a simple path.

**Lemma 3.33.** If $|m| > |n|$ and $\mathcal{G}$ is an $(m, n)$-graph with a vertex of finite label, then there is a vertex $v \in V(\mathcal{G})$ such that $\deg_{\text{in}}(v) < |m|$.

*Proof.* Assume by contradiction that $\deg_{\text{in}}(v) = |m|$ for all $v \in V(\mathcal{G})$. Then we can build inductively an infinite path $(e_k)_{k \in \mathbb{N}}$ consisting of negative edges with $L(\mathbf{s}(e_0))$ finite. By the previous lemma this path goes through some vertex $v_0$ that $|L(v_0)|_p < |m|_p$. Then $\deg_{\text{in}}(v_0) = \gcd(L(v_0), m) < |m|$, a contradiction. $\square$

## 3.6 Bass-Serre graphs and Bass-Serre theory

Take $m, n \in \mathbb{Z}\setminus\{0\}$. Set $\Gamma := \text{BS}(m, n) = \langle b, t | tb^m t^{-1} = b^n \rangle$ and put $S := \{b, t\}$. Denote by $\mathcal{T}$ the associated Bass-Serre tree and remark that it is the
It is easy to check that all $\alpha_f$ are actions of BS(2, 2) whose Bass-Serre graph is $\mathcal{G}$, that $\alpha_f$ and $\alpha_g$ are non-conjugate for $f \neq g$, and that there are continuum many such actions.

## 3.5 Additional properties of $(m, n)$-graphs

In this section, we collect some basic consequences of the definition of $(m, n)$-graphs. Observe that Equation (3.13) is equivalent to the fact that

$$ \max(|L(s(e))|_p - |n|_p, 0) = |L(e)|_p = \max(|L(t(e))|_p - |m|_p, 0) \quad (3.25) $$

from which we obtain the following.

**Remark 3.26.** Consider an oriented labeled graph $\mathcal{G} = (V, E)$ with label map $L: V \sqcup E \to \mathbb{Z}_{\ge 1}$ satisfying $L(\bar{e}) = L(e)$ for every edge $e$. The labeled graph $\mathcal{G}$ is an $(m, n)$-graph if and only if the following two conditions hold:

* for every positive edge $e$ and every prime $p$ such that $|L(e)|_p \ge 1$,

$$ |L(\mathbf{s}(e))|_p = |L(e)|_p + |n|_p \quad \text{and} \quad |L(\mathbf{t}(e))|_p = |L(e)|_p + |m|_p, \quad (3.27) $$

* for every positive edge $e$ and every prime $p$ such that $|L(e)|_p = 0$,

$$ 0 \le |L(\mathbf{s}(e))|_p \le |n|_p \quad \text{and} \quad 0 \le |L(\mathbf{t}(e))|_p \le |m|_p. \quad (3.28) $$

In particular, in an $(m, n)$-graph, $|L(\mathbf{s}(e))|_p > |n|_p$ if and only if $|L(\mathbf{t}(e))|_p > |m|_p$, and if one of these two equivalent conditions is met then

$$ |L(\mathbf{t}(e))|_p = |L(\mathbf{s}(e))|_p + |m|_p - |n|_p. \quad (3.29) $$

**Lemma 3.30.** Fix a prime $p$ such that $|n|_p < |m|_p$ and let $\mathcal{G}$ be an $(m, n)$-graph. If $(e_k)_{k \ge 1}$ is any infinite path consisting of positive edges with $L(\mathbf{s}(e_1)) \ne \infty$ and $|L(\mathbf{s}(e_1))|_p > |n|_p$, then

$$ \lim_{k \to +\infty} |L(\mathbf{s}(e_k))|_p = +\infty. $$

If $(e_k)_{k \ge 1}$ is any infinite path consisting of negative edges with $L(\mathbf{s}(e_1)) \ne \infty$, then

$$ \limsup_{k \to +\infty} |L(\mathbf{s}(e_k))|_p < |m|_p. $$
underlying oriented graph of the Bass-Serre graph of the transitive and free action: $\mathcal{T} = \mathbf{BS}(\Gamma \hookrightarrow \Gamma)$.

Besides the Schreier graph, we can associate to each subgroup $\Lambda \leq \Gamma$ two decorated graphs:

* the Bass-Serre graph of the action $\Lambda \setminus \Gamma \hookrightarrow \Gamma$;

* the quotient graph of groups $\Lambda \setminus \mathcal{T}$ of the action $\Lambda \hookrightarrow \mathcal{T}$.

Let us observe that the underlying oriented graphs of the two above decorated graphs are the same. Indeed they are obtained as quotients of commuting actions as one can see in the following diagram where by $\hookrightarrow^V \langle b \rangle$ we mean that $\langle b \rangle$ acts only on the set of vertices, where the $\perp$ arrows are graph morphisms obtained by quotienting by left $\Lambda$-actions, and where the dashed $\searrow$ arrows are projections as in Definition 3.10:

<figure><img src="image_7.png" /></figure>

Next, observe that, $\mathbf{BS}(\Lambda \setminus \Gamma \hookrightarrow \Gamma)$ being saturated, one has $\deg_{\text{in}}(v) = \text{gcd}(L(v), m)$ and hence, for every edge $\deg_{\text{out}}(v) = \text{gcd}(L(v), n)$ for every vertex $v$ in this graph. Hence, for every edge $e$, one has

$$ \frac{L(\mathbf{s}(e))}{L(e)} = \text{gcd}(L(\mathbf{s}(e)), n) = \deg_{\text{out}}(\mathbf{s}(e)) \quad \text{and} \quad \frac{L(\mathbf{t}(e))}{L(e)} = \deg_{\text{in}}(\mathbf{t}(e)), $$

so that Remark 2.8 and Proposition 2.9 can be reformulated in terms of the labels of the Bass-Serre graph $\mathbf{BS}(\Lambda \setminus \Gamma \hookrightarrow \Gamma)$.

**Proposition 3.34.** Let $m$ and $n$ be non-zero integers. Let $\mathcal{G}$ be a saturated connected $(m, n)$-graph and let $\Lambda$ be a subgroup of $\Gamma = \mathbf{BS}(m, n)$ such that $\mathbf{BS}(\Lambda \setminus \Gamma \hookrightarrow \Gamma) \simeq \mathcal{G}$.
Proof. If $(e_k)_{k \ge 1}$ is an infinite path consisting of positive edges such that $|L(\mathbf{s}(e_1))|_p > |n|_p$, then by a straightforward induction using Equation (3.29) we have that

$$|L(\mathbf{s}(e_k))|_p = |L(\mathbf{s}(e_1))|_p + k(|m|_p - |n|_p) \quad (3.31)$$

for all $k \ge 1$. The first result follows.

For the second one, let $(e_k)_{k \ge 1}$ be an infinite path consisting of negative edges. By exchanging the roles in Equation (3.29), we have the claim: if $e$ is a negative edge then $|L(\mathbf{s}(e))|_p > |m|_p$ if and only if $|L(\mathbf{t}(e))|_p > |n|_p$; and when this occurs $|L(\mathbf{t}(e))|_p = |L(\mathbf{s}(e))|_p - |m|_p + |n|_p$. Thus, $|L(\mathbf{s}(e_{k+1}))|_p = |L(\mathbf{t}(e_k))|_p < |L(\mathbf{s}(e_k))|_p$ as long as $|L(\mathbf{s}(e_k))|_p > |m|_p$. So there must be $k_0 \in \mathbb{N}$ such that $|L(\mathbf{s}(e_{k_0}))|_p \le |m|_p$ (this could have already happened for $k_0 = 1$). From this point, we have $|L(\mathbf{s}(e_{k_0+1}))|_p = |L(\mathbf{t}(e_{k_0}))|_p \le |n|_p < |m|_p$ and an inductive use of the claim gives $|L(\mathbf{s}(e_k))|_p \le |n|_p < |m|_p$ for all $k > k_0$. This finishes the proof. $\square$

**Remark 3.32.** It follows from Equation (3.31) that any infinite path $(e_k)_{k \ge 1}$ consisting of positive edges with $L(\mathbf{s}(e_1)) \ne \infty$ and $|L(\mathbf{s}(e_1))|_p > |n|_p$ has to be a simple path.

**Lemma 3.33.** If $|m| > |n|$ and $\mathcal{G}$ is an $(m, n)$-graph with a vertex of finite label, then there is a vertex $v \in V(\mathcal{G})$ such that $\deg_{\text{in}}(v) < |m|$.

Proof. Assume by contradiction that $\deg_{\text{in}}(v) = |m|$ for all $v \in V(\mathcal{G})$. Then we can build inductively an infinite path $(e_k)_{k \in \mathbb{N}}$ consisting of negative edges with $L(\mathbf{s}(e_0))$ finite. By the previous lemma this path goes through some vertex $v_0$ that $|L(v_0)|_p < |m|_p$. Then $\deg_{\text{in}}(v_0) = \gcd(L(v_0), m) < |m|$, a contradiction. $\square$

## 3.6 Bass-Serre theory for $\text{BS}(m, n)$

Take $m, n \in \mathbb{Z} \setminus \{0\}$. Set $\Gamma := \text{BS}(m, n) = \langle b, t | tb^m t^{-1} = b^n \rangle$ and put $S := \{b, t\}$. Denote by $\mathcal{T}$ the associated Bass-Serre tree and remark that it is the underlying oriented graph of the Bass-Serre graph of the transitive and free action: $\mathcal{T} = \text{BS}(\Gamma \hookrightarrow \Gamma)$.

Besides the Schreier graph, we can associate to each subgroup $\Lambda \le \Gamma$ two decorated graphs:

* the Bass-Serre graph of the action $\Lambda \setminus \Gamma \hookrightarrow \Gamma$;

* the quotient graph of groups $\Lambda \setminus \mathcal{T}$ of the action $\Lambda \hookrightarrow \mathcal{T}$.
1. If all labels of $\mathcal{G}$ are infinite, then $\Lambda$ is a free group, namely isomorphic to the fundamental group of the graph $\mathcal{G}$.

2. If all labels of $\mathcal{G}$ are finite, then the quotient graph of groups arising from the action $\Lambda \to \mathcal{T}$ is isomorphic to the graph of groups obtained by attaching a copy of $\mathbb{Z}$ to every vertex and every edge of $\mathcal{G}$, with structural maps of positive edges

$$ \mathbb{Z}_e \hookrightarrow \mathbb{Z}_{\mathbf{s}(e)}, \quad k \mapsto \frac{n \cdot L(e)}{L(\mathbf{s}(e))} \cdot k, $$

$$ \mathbb{Z}_e \hookrightarrow \mathbb{Z}_{\mathbf{t}(e)}, \quad k \mapsto \frac{m \cdot L(e)}{L(\mathbf{t}(e))} \cdot k. $$

In particular, combining Proposition 3.34 and Lemma 3.33 we get a corollary equivalent to Corollary 2.15.

**Corollary 3.35.** Let $m$ and $n$ be non-zero integers such that $|m| \neq |n|$. Then the isomorphism type of $\Lambda \leq BS(m, n)$ depends only on the graph structure of $BS(\Lambda)$.

*Proof.* Recall that if an $(m, n)$-graph is saturated and has only infinite labels, then all vertices have ingoing degree $|m|$ and outgoing degree $|n|$. Lemma 3.33 thus allows us to detect whether the Bass Serre graph of $\Lambda$ contains infinite labels by purely looking at its graph structure: it has infinite labels if and only if all vertices have degree $|n| + |m|$. The result now follows from Proposition 3.34. $\square$

**Remark 3.36.** When $|m| = |n|$, the statement analogue to that of Corollary 3.35 fails since the central subgroup $\Lambda = \langle b^{2n} \rangle$ has the same Bass-Serre graph as the trivial subgroup $\{\text{id}\}$.

# 4 Phenotype

In this section, we introduce a central invariant to understand transitive $BS(m, n)$-(pre)-actions: the *phenotype* (see Definition 4.9). We first define the $(m, n)$-phenotype of a natural number. We then prove that given a transitive pre-action $(\tau, \beta)$, all cardinalities of $\beta$-orbits have the same phenotype.
Let us observe that the underlying oriented graphs of the two above decorated
graphs are the same. Indeed they are obtained as quotients of commuting
actions as one can see in the following diagram where by $\leftarrow^V \langle b \rangle$ we mean
that $\langle b \rangle$ acts only on the set of vertices, where the $\downarrow$ arrows are graph
morphisms obtained by quotienting by left $\Lambda$-actions, and where the dashed
$\rightarrow$ arrows are projections as in Definition 3.10:

<figure><img src="image_5.png" /></figure>

Next, observe that, $\mathbf{BS}(\Lambda\setminus\Gamma \leftarrow \Gamma)$ being saturated, one has $\deg_{\text{in}}(v) = \gcd(L(v), m)$ and $\deg_{\text{out}}(v) = \gcd(L(v), n)$ for every vertex $v$ in this graph.
Hence, for every edge $e$, one has

$$ \frac{L(\mathbf{s}(e))}{L(e)} = \gcd(L(\mathbf{s}(e)), n) = \deg_{\text{out}}(\mathbf{s}(e)) \quad \text{and} \quad \frac{L(\mathbf{t}(e))}{L(e)} = \deg_{\text{in}}(\mathbf{t}(e)). $$

Thus Remark 2.8 and Proposition 2.9 can be immediately reformulated in
terms of the labels of the Bass-Serre graph $\mathbf{BS}(\Lambda\setminus\Gamma \leftarrow \Gamma)$ as follows:

**Proposition 3.34.** Let $m$ and $n$ be non-zero integers. Let $\mathcal{G}$ be a saturated connected $(m, n)$-graph and let $\Lambda$ be a subgroup of $\Gamma = \mathrm{BS}(m, n)$ such that $\mathbf{BS}(\Lambda\setminus\Gamma \leftarrow \Gamma) \simeq \mathcal{G}$.

1. If all labels of $\mathcal{G}$ are infinite, then $\Lambda$ is a free group, namely isomorphic to the fundamental group of the graph $\mathcal{G}$.

2. If all labels of $\mathcal{G}$ are finite, then the quotient graph of groups arising from the action $\Lambda \leftarrow \mathcal{T}$ is isomorphic to the graph of groups obtained by attaching a copy of $\mathbb{Z}$ to every vertex and every edge of $\mathcal{G}$, with
## 4.1 Phenotypes of natural numbers

Recall that $\mathcal{P}$ denotes the set of prime numbers and that given $p \in \mathcal{P}$ and $k \in \mathbb{Z}$, we denote by $|k|_p$ the $p$-adic valuation of $k$.

**Definition 4.1 (Phenotype of a natural number).** Let $k \in \mathbb{Z}_{\ge 1}$. We set

$$ \mathcal{P}_{m,n} := \{p \in \mathcal{P} : |m|_p = |n|_p\}, $$

$$ \mathcal{P}_{m,n}(k) := \{p \in \mathcal{P} : |m|_p = |n|_p \text{ and } |k|_p > |n|_p\}. $$

The $(m, n)$-phenotype of $k$, denoted by $\text{Ph}_{m,n}(k)$, is the following positive integer:

$$ \text{Ph}_{m,n}(k) := \prod_{p \in \mathcal{P}_{m,n}(k)} p^{|k|_p}. $$

If $k = \infty$, we set $\text{Ph}_{m,n}(k) := \infty$.

**Example 4.2.** If $m$ and $n$ are coprime, then for every $k \in \mathbb{Z}$

$$ \mathcal{P}_{m,n} = \{p \in \mathcal{P} : p \text{ does not divide } mn\} $$

$$ \mathcal{P}_{m,n}(k) = \{p \in \mathcal{P} : p \text{ divides } k \text{ and } p \text{ does not divide } mn\}. $$

In this case, $\text{Ph}_{m,n}(k)$ is the greater divisor of $k$ that is coprime to $mn$.

**Example 4.3.** If $m = 2^2 \cdot 3^2 \cdot 5$ and $n = 2^2 \cdot 3$, then $\mathcal{P}_{m,n} = \mathcal{P}\setminus\{3, 5\}$ and

$$ \mathcal{P}_{m,n}(k) = \begin{cases} \{p \in \mathcal{P} : p \text{ divides } k\} \setminus \{2, 3, 5\} & \text{if } 2^3 \text{ does not divide } k \\ \{p \in \mathcal{P} : p \text{ divides } k\} \setminus \{3, 5\} & \text{if } 2^3 \text{ divides } k. \end{cases} $$

For example $\text{Ph}_{m,n}(2 \cdot 3 \cdot 7) = 7$ and $\text{Ph}_{m,n}(2^5 \cdot 3 \cdot 7) = 2^5 \cdot 7$.

**Remark 4.4.** If $k, l$ both have phenotype $q$, then so do their $\text{lcm}$ and $\text{gcd}$.

The following lemma will be useful in Section 5.

**Lemma 4.5.** Let $q = \text{Ph}_{m,n}(k)$ be a finite $(m, n)$-phenotype. Then $\text{Ph}_{m,n}^{-1}(\{q\})$ is finite if and only if $|m| = |n|$.

*Proof.* Assume first $|m| \neq |n|$. In this case, there is a prime number $p$ such that $|m|_p \neq |n|_p$. We get $\text{Ph}_{m,n}(p^i k) = q$ for all $i$, hence $\text{Ph}_{m,n}^{-1}(\{q\})$ is infinite.

If $|m| = |n|$, then $\mathcal{P}_{m,n} = \mathcal{P}$. If $k$ and $k'$ are two integers with the same phenotype, the only primes $p$ for which the valuations of $k$ and $k'$ may differ are those for which $|k|_p \le |m|_p$ and in this case $|k'|_p$ must also be bounded by $|m|_p$. There are only finitely many such $k'$. $\square$
structural maps of positive edges

$$ \mathbb{Z}_e \hookrightarrow \mathbb{Z}_{\mathfrak{s}(e)}, \quad k \mapsto \frac{n \cdot L(e)}{L(\mathfrak{s}(e))} \cdot k, $$

$$ \mathbb{Z}_e \hookrightarrow \mathbb{Z}_{\mathfrak{t}(e)}, \quad k \mapsto \frac{m \cdot L(e)}{L(\mathfrak{t}(e))} \cdot k. $$

Then, combining Proposition 3.34 and Lemma 3.33, we get the following rephrasing of Corollary 2.15:

**Corollary 3.35.** Let $m$ and $n$ be non-zero integers such that $|m| \neq |n|$. Then the isomorphism type of $\Lambda \leq BS(m, n)$ depends only on the graph structure of $BS(\Lambda)$.

*Proof.* Recall that if an $(m, n)$-graph is saturated and has only infinite labels, then all vertices have incoming degree $|m|$ and outgoing degree $|n|$. Lemma 3.33 thus allows us to detect whether the Bass Serre graph of $\Lambda$ contains infinite labels by purely looking at its graph structure: it has infinite labels if and only if all vertices have degree $|n| + |m|$. The result now follows from Proposition 3.34. $\square$

**Remark 3.36.** When $|m| = |n|$, the statement analogue to that of Corollary 3.35 fails since the central subgroup $\Lambda = \langle b^{2n} \rangle$ has the same Bass-Serre graph as the trivial subgroup $\{\text{id}\}$.

# 4 Phenotype

In this section, we introduce a central invariant to understand transitive $BS(m, n)$-(pre)-actions: the *phenotype* (see Definition 4.9). We first define the $(m, n)$-phenotype of a natural number. We then prove that given a transitive pre-action $(\tau, \beta)$, all cardinalities of $\beta$-orbits have the same phenotype.

## 4.1 Phenotypes of natural numbers

Recall that $\mathcal{P}$ denotes the set of prime numbers and that given $p \in \mathcal{P}$ and $k \in \mathbb{Z}$, we denote by $|k|_p$ the $p$-adic valuation of $k$.
## 4.2 Phenotypes of $(m, n)$-graphs

If $v$ is a vertex of an $(m, n)$-graph, we use the shorter expression “phenotype of the vertex $v$” to mean “phenotype of the label of the vertex $v$”. The key feature of the notion of phenotype is the following statement.

**Proposition 4.6.** The vertices of a connected $(m, n)$-graph all have the same $(m, n)$-phenotype.

*Proof.* It is enough to check that for any positive edge $e$ from $v_-$ to $v_+$, the phenotypes of $v_-$ and $v_+$ are the same. If the phenotype of one of them is infinite, then this is a direct consequence of Equation (3.13) from Definition 3.12. Otherwise, remark that for every positive integer $k$ and every $p \in \mathcal{P}_{m,n}$,

$$ \left| \frac{k}{\gcd(k, n)} \right|_p > 0 \Leftrightarrow p \in \mathcal{P}_{m,n}(k). $$

Equation (3.13) implies

$$ \left| \frac{L(v_-)}{\gcd(L(v_-), n)} \right|_p = |L(e)|_p = \left| \frac{L(v_+)}{\gcd(L(v_+), m)} \right|_p $$

and hence $\mathcal{P}_{m,n}(L(v_-)) = \mathcal{P}_{m,n}(L(v_+))$. If $p \in \mathcal{P}_{m,n}(L(v_-))$, then $L(v_-)$ has higher $p$-valuation than $m$ and $n$, so that

$$ |L(v_-)|_p - |n|_p = \left| \frac{L(v_-)}{\gcd(L(v_-), n)} \right|_p = \left| \frac{L(v_+)}{\gcd(L(v_+), m)} \right|_p = |L(v_+)|_p - |m|_p. $$

Since $|n|_p = |m|_p$, we conclude that $|L(v_-)|_p = |L(v_+)|_p$ for all $p \in \mathcal{P}_{m,n}(L(v_-)) = \mathcal{P}_{m,n}(L(v_+))$. Therefore $L(v_-)$ and $L(v_+)$ share the same phenotype. $\square$

**Remark 4.7.** One can prove that the edges of a connected $(m, n)$-graph also all have the same $(m, n)$-phenotype. However, it is a coarser invariant: there are connected graphs with different vertex phenotypes, but with the same edge phenotype. For example, fix

$$ m = 2^2 \cdot 3^2 \cdot 5, \quad n = 2^2 \cdot 3 $$

and consider the graph consisting of a single oriented edge $e$ and its two endpoints. If the label of its origin is $L(\mathbf{s}(e)) = 2^3 \cdot 7$, then

$$ L(e) = \frac{L(\mathbf{s}(e))}{\gcd(L(\mathbf{s}(e)), n)} = 2 \cdot 7 \text{ and } \mathrm{Ph}_{m,n}(L(e)) = 7 $$
**Definition 4.1** (Phenotype of a natural number). Let $k \in \mathbb{Z}_{\ge 1}$. We set

$$ \mathcal{P}_{m,n} := \{p \in \mathcal{P} : |m|_p = |n|_p\}, $$

$$ \mathcal{P}_{m,n}(k) := \{p \in \mathcal{P} : |m|_p = |n|_p \text{ and } |k|_p > |n|_p\}. $$

The $(m, n)$-phenotype of $k$, denoted by $\text{Ph}_{m,n}(k)$, is the following positive integer:

$$ \text{Ph}_{m,n}(k) := \prod_{p \in \mathcal{P}_{m,n}(k)} p^{|k|_p}. $$

If $k = \infty$, we set $\text{Ph}_{m,n}(k) := \infty$.

**Example 4.2.** If $m$ and $n$ are coprime, then for every $k \in \mathbb{Z}$

$$ \mathcal{P}_{m,n} = \{p \in \mathcal{P} : p \text{ does not divide } mn\} $$

$$ \mathcal{P}_{m,n}(k) = \{p \in \mathcal{P} : p \text{ divides } k \text{ and } p \text{ does not divide } mn\}. $$

In this case, $\text{Ph}_{m,n}(k)$ is the greatest divisor of $k$ that is coprime to $mn$.

**Example 4.3.** If $m = 2^2 \cdot 3^2 \cdot 5$ and $n = 2^2 \cdot 3$, then $\mathcal{P}_{m,n} = \mathcal{P} \setminus \{3, 5\}$ and

$$ \mathcal{P}_{m,n}(k) = \begin{cases} \{p \in \mathcal{P} : p \text{ divides } k\} \setminus \{2, 3, 5\} & \text{if } 2^3 \text{ does not divide } k \\ \{p \in \mathcal{P} : p \text{ divides } k\} \setminus \{3, 5\} & \text{if } 2^3 \text{ divides } k. \end{cases} $$

For example $\text{Ph}_{m,n}(2 \cdot 3 \cdot 7) = 7$ and $\text{Ph}_{m,n}(2^5 \cdot 3 \cdot 7) = 2^5 \cdot 7$.

**Remark 4.4.** If $k, l$ both have phenotype $q$, then so do their lcm and gcd.

The following lemma will be useful in Section 5.

**Lemma 4.5.** Let $q = \text{Ph}_{m,n}(k)$ be a finite $(m, n)$-phenotype. Then $\text{Ph}_{m,n}^{-1}(\{q\})$ is finite if and only if $|m| = |n|$.

*Proof.* Assume first $|m| \neq |n|$. In this case, there is a prime number $p$ such that $|m|_p \neq |n|_p$. We get $\text{Ph}_{m,n}(p^i k) = q$ for all $i$, hence $\text{Ph}_{m,n}^{-1}(\{q\})$ is infinite.

If $|m| = |n|$, then $\mathcal{P}_{m,n} = \mathcal{P}$. If $k$ and $k'$ are two integers with the same phenotype, the only primes $p$ for which the valuations of $k$ and $k'$ may differ are those for which $|k|_p \le |m|_p$ and in this case $|k'|_p$ must also be bounded by $|m|_p$. There are only finitely many such $k'$. $\square$
while $\mathrm{Ph}_{m,n}(L(\mathbf{s}(e))) = 2^3 \cdot 7$. If instead we set the label of its origin to be $L(\mathbf{s}(e)) = 2^4 \cdot 7$, then we get

$$L(e) = 2^2 \cdot 7 \text{ and } \mathrm{Ph}(L(e)) = 7$$

while $\mathrm{Ph}_{m,n}(L(\mathbf{s}(e))) = 2^4 \cdot 7 \neq 2^3 \cdot 7$. We will thus not use the phenotype of edges.

Proposition 4.6 allows us to define the phenotypes of connected $(m, n)$-graphs and transitive $\mathrm{BS}(m, n)$-pre-actions.

**Definition 4.8.** The **phenotype of a connected $(m, n)$-graph $\mathcal{G}$** is the common phenotype of the labels of its vertices. We denote it $\mathrm{Ph}(\mathcal{G})$.

## 4.3 Phenotypes of $\mathrm{BS}(m, n)$-actions

Recall that a pre-action is transitive if its Schreier graph is connected, which is equivalent to its Bass-Serre graph being connected.

**Definition 4.9.** The **phenotype of a transitive (pre)-action $\alpha$ of $\mathrm{BS}(m, n)$** is the common phenotype of the cardinalities $\mathrm{Ph}_{m,n}(|x \langle b \rangle|)$ of its $\langle b \rangle$-orbits. We denote it $\mathrm{Ph}(\alpha)$.

By definition, the phenotype of any transitive (pre)-action coincides with the phenotype of its Bass-Serre graph.

**Remark 4.10.** Any $\mathrm{BS}(m, n)$-action with finite Bass-Serre graph and finite phenotype is necessarily an action on a finite set whose cardinality is the sum of the labels of the vertices.

For infinite phenotype, we have the following.

**Lemma 4.11.** There exists an infinite phenotype transitive $\mathrm{BS}(m, n)$-action with finite Bass-Serre graph if and only if $|m| = |n|$.

*Proof.* Consider an infinite phenotype $\mathrm{BS}(m, n)$-action with finite Bass-Serre graph $\mathcal{G}$. Since $\mathcal{G}$ is saturated, all its vertices have outgoing degree $|n|$ and incoming degree $|m|$. But there must be globally as many outgoing edges as incoming edges, so since $\mathcal{G}$ is finite we must have $|n| = |m|$.

Conversely if $|n| = |m|$, consider the bouquet of $|n|$ circles with edges and vertices labeled by $\infty$, and observe that this is a connected saturated $(m, n)$-graph. Proposition 3.22 provides a transitive action having this labeled bouquet of circles as its finite Bass-Serre graph of infinite phenotype. $\square$
## 4.2 Phenotypes of $(m, n)$-graphs

If $v$ is a vertex of an $(m, n)$-graph, we use the shorter expression “phenotype of the vertex $v$” to mean “phenotype of the label of the vertex $v$”. The key feature of the notion of phenotype is the following statement.

**Proposition 4.6.** *The vertices of a connected $(m, n)$-graph all have the same $(m, n)$-phenotype.*

*Proof.* It is enough to check that for any positive edge $e$ from $v_-$ to $v_+$, the phenotypes of $v_-$ and $v_+$ are the same. If the phenotype of one of them is infinite, then this is a direct consequence of Equation (3.13) from Definition 3.12. Otherwise, remark that for every positive integer $k$ and every $p \in \mathcal{P}_{m,n}$,

$$ \left| \frac{k}{\gcd(k, n)} \right|_p > 0 \Leftrightarrow p \in \mathcal{P}_{m,n}(k). $$

Equation (3.13) implies

$$ \left| \frac{L(v_-)}{\gcd(L(v_-), n)} \right|_p = |L(e)|_p = \left| \frac{L(v_+)}{\gcd(L(v_+), m)} \right|_p $$

and hence $\mathcal{P}_{m,n}(L(v_-)) = \mathcal{P}_{m,n}(L(v_+))$. If $p \in \mathcal{P}_{m,n}(L(v_-))$, then $L(v_-)$ has higher $p$-valuation than $m$ and $n$, so

$$ |L(v_-)|_p - |n|_p = \left| \frac{L(v_-)}{\gcd(L(v_-), n)} \right|_p = \left| \frac{L(v_+)}{\gcd(L(v_+), m)} \right|_p = |L(v_+)|_p - |m|_p. $$

Since $|n|_p = |m|_p$, we conclude that $|L(v_-)|_p = |L(v_+)|_p$ for all $p \in \mathcal{P}_{m,n}(L(v_-)) = \mathcal{P}_{m,n}(L(v_+))$. Therefore $L(v_-)$ and $L(v_+)$ share the same phenotype. $\square$

**Remark 4.7.** One can prove that the edges of a connected $(m, n)$-graph also all have the same $(m, n)$-phenotype. However, it is a coarser invariant: there are connected graphs with different vertex phenotypes, but with the same edge phenotype. For example, fix

$$ m = 2^2 \cdot 3^2 \cdot 5, \quad n = 2^2 \cdot 3 $$

and consider the graph consisting of a single oriented edge $e$ and its two endpoints. If the label of its origin is $L(\mathbf{s}(e)) = 2^3 \cdot 7$, then

$$ L(e) = \frac{L(\mathbf{s}(e))}{\gcd(L(\mathbf{s}(e)), n)} = 2 \cdot 7 \text{ and } \mathrm{Ph}_{m,n}(L(e)) = 7 $$
## 4.4 Merging pre-actions

In order to establish some of the main results of this article, we will need “cut and paste” operations on pre-actions, for instance:

* putting two prescribed pre-actions inside a single transitive action (useful for topological transitivity properties);

* modifying an action so as to add or remove a circuit in its Schreier graph (useful to get a new action that is close but distinct from the original one).

We now present these “cut and paste” operations. The main one is the following and the rest of this section will be devoted to its proof. Other useful results will appear in the course of the proof.

**Theorem 4.12 (The merging machine).** Assume $|m| \ge 2$ and $|n| \ge 2$. Let $\alpha_1$ and $\alpha_2$ two transitive non-saturated pre-actions of $\text{BS}(m, n)$ with the same phenotype. There exists a transitive action $\alpha$ which contains copies of $\alpha_1$ and $\alpha_2$ with disjoint domains.

Given a pre-action $\alpha = (\beta, \tau)$ and two sub-pre-actions $\alpha_1, \alpha_2$, let us recall that the domain of $\alpha$ is the set $\text{dom}(\beta) = \text{rang}(\beta)$. Notice that $\alpha_1$ and $\alpha_2$ have disjoint domains if and only if their Bass-Serre graphs $\text{BS}(\alpha_1)$ and $\text{BS}(\alpha_2)$ are disjoint (that is, have no common vertex) in $\text{BS}(\alpha)$.

First, taking advantage of Proposition 3.23, we reduce to the case of $(m, n)$-graphs, for which the analogous result is the following.

**Theorem 4.13 (The merging machine for $(m, n)$-graphs).** Assume $|m| \ge 2$ and $|n| \ge 2$. Let $\mathcal{G}_1$ and $\mathcal{G}_2$ be two connected and non-saturated $(m, n)$-graphs with the same phenotype. There exists a connected and saturated $(m, n)$-graph $\mathcal{G}$ which contains disjoint copies of $\mathcal{G}_1$ and $\mathcal{G}_2$.

**Remark 4.14.** The hypothesis that both $|m|, |n| \ge 2$ is necessary. If $m = 1$ but $|n| \ne 1$, we can consider the $(1, n)$-graph consisting of a single vertex with infinite label and only one loop. This graph is not saturated but it cannot be connected to another copy of itself. Indeed, the reader can check that the only saturated graph containing it admits a unique circuit, namely the loop itself.

*Proof of Theorem 4.12 based on Theorem 4.13.* The Bass-Serre graphs $\text{BS}(\alpha_1)$ and $\text{BS}(\alpha_2)$ are connected non-saturated $(m, n)$-graphs with the same phenotype. Therefore we can apply Theorem 4.13 to obtain a connected and saturated $(m, n)$-graph $\mathcal{G}$ which contains disjoint copies of $\text{BS}(\alpha_1)$ and $\text{BS}(\alpha_2)$.
while $\mathrm{Ph}_{m,n}(L(\mathbf{s}(e))) = 2^3 \cdot 7$. If instead we set the label of its origin to be $L(\mathbf{s}(e)) = 2^4 \cdot 7$, then we get

$$L(e) = 2^2 \cdot 7 \text{ and } \mathrm{Ph}(L(e)) = 7$$

while $\mathrm{Ph}_{m,n}(L(\mathbf{s}(e))) = 2^4 \cdot 7 \neq 2^3 \cdot 7$. We will thus not use the phenotype of edges.

Proposition 4.6 allows us to define the phenotypes of connected $(m, n)$-graphs and transitive $\mathrm{BS}(m, n)$-pre-actions.

**Definition 4.8.** The **phenotype of a connected $(m, n)$-graph $\mathcal{G}$** is the common phenotype of the labels of its vertices. We denote it $\mathrm{Ph}(\mathcal{G})$.

## 4.3 Phenotypes of $\mathrm{BS}(m, n)$-actions

Recall that a pre-action is transitive if its Schreier graph is connected, which is equivalent to its Bass-Serre graph being connected.

**Definition 4.9.** The **phenotype of a transitive (pre)-action $\alpha$ of $\mathrm{BS}(m, n)$** is the common phenotype of the cardinalities $\mathrm{Ph}_{m,n}(|x\langle b\rangle|)$ of its $\langle b\rangle$-orbits. We denote it $\mathrm{Ph}(\alpha)$.

By definition, the phenotype of any transitive (pre)-action coincides with the phenotype of its Bass-Serre graph.

**Remark 4.10.** Any $\mathrm{BS}(m, n)$-action with finite Bass-Serre graph and finite phenotype is necessarily an action on a finite set whose cardinality is the sum of the labels of the vertices.

For infinite phenotype, we have the following.

**Lemma 4.11.** There exists an infinite phenotype transitive $\mathrm{BS}(m, n)$-action with finite Bass-Serre graph if and only if $|m| = |n|$.

*Proof.* Consider an infinite phenotype $\mathrm{BS}(m, n)$-action with finite Bass-Serre graph $\mathcal{G}$. Since $\mathcal{G}$ is saturated, all its vertices have outgoing degree $|n|$ and incoming degree $|m|$. But there must be globally as many outgoing edges as incoming edges, so since $\mathcal{G}$ is finite we must have $|n| = |m|$.

Conversely if $|n| = |m|$, consider the bouquet of $|n|$ circles with edges and vertices labeled by $\infty$, and observe that this is a connected saturated $(m, n)$-graph. Proposition 3.22 provides a transitive action having this labeled bouquet of circles as its finite Bass-Serre graph of infinite phenotype. $\square$
Then, we apply Proposition 3.23 to the pre-action $\alpha_1 \sqcup \alpha_2$, whose Bass-Serre graph $\mathbf{BS}(\alpha_1) \sqcup \mathbf{BS}(\alpha_2)$ is contained in $\mathcal{G}$, to ensure the existence of a $\mathbf{BS}(m, n)$-pre-action $\alpha$ which extends $\alpha_1 \sqcup \alpha_2$. Thus $\alpha$ extends both $\alpha_1$ and $\alpha_2$ with disjoint domains. Since $\mathcal{G}$ is connected and saturated, $\alpha$ is a transitive and saturated pre-action, i.e., it is a genuine transitive action of $\mathbf{BS}(m, n)$ that satisfies the requirements of Theorem 4.12. $\square$

We now present some general results we will use in order to prove Theorem 4.13. We begin with two easy properties of phenotypes which will be useful in the proof.

**Lemma 4.15.** For any $k \in \mathbb{Z}_{\ge 1}$, if $q = \mathrm{Ph}_{m,n}(k)$, then $\mathrm{Ph}_{m,n}(q) = q$ and $\mathrm{gcd}(q, n) = \mathrm{gcd}(q, m)$.

*Proof.* We get directly from Definition 4.1 that $|q|_p = |k|_p$ if $p \in \mathcal{P}_{m,n}(k)$, and $|q|_p = 0$ for the other primes $p$. Consequently, we get $\mathcal{P}_{m,n}(q) = \mathcal{P}_{m,n}(k)$ and then $\mathrm{Ph}_{m,n}(q) = \mathrm{Ph}_{m,n}(k) = q$. Finally, since every prime $p$ dividing $q$ satisfies $|m|_p = |n|_p$ and $|n|_p < |q|_p$, we obtain

$$ \mathrm{gcd}(q, n) = \prod_{p \in \mathcal{P}: p|q} p^{|n|_p} = \prod_{p \in \mathcal{P}: p|q} p^{|m|_p} = \mathrm{gcd}(q, m). \quad \square $$

In the following lemma, by welding two vertices we mean taking the quotient graph obtained by identifying these vertices. Its proof is a direct consequence of the definition of an $(m, n)$-graph, so we omit it.

**Lemma 4.16 (Welding lemma).** Let $m, n \in \mathbb{Z} \setminus \{0\}$ and let $\mathcal{G}$ be an $(m, n)$-graph and $v, w$ be two distinct vertices such that:

* $L := L(v) = L(w)$;

* $\mathrm{deg}_{\mathrm{out}}(v) + \mathrm{deg}_{\mathrm{out}}(w) \le \mathrm{gcd}(n, L)$;

* $\mathrm{deg}_{\mathrm{in}}(v) + \mathrm{deg}_{\mathrm{in}}(w) \le \mathrm{gcd}(m, L)$.

*Welding together $v$ and $w$ delivers an $(m, n)$-graph.* $\square$

Note that in this lemma $\mathcal{G}$ can be finite or infinite, connected or not. Together with the welding lemma, the following result will allow us to connect non saturated $(m, n)$-graphs via the well-known technique of arc welding.

**Theorem 4.17 (Connecting lemma).** Assume $|m| \ge 2$ and $|n| \ge 2$. Let $k, \ell \in \mathbb{Z}_{\ge 1}$ such that $\mathrm{Ph}_{m,n}(k) = \mathrm{Ph}_{m,n}(\ell)$, and let $\varepsilon_k, \varepsilon_\ell \in \{+, -\}$. There exists a $(m, n)$-graph $\mathcal{G}$ which is a simple edge path $(e_1, \dots, e_h)$ of length $h \ge 1$ such that:
## 4.4 Merging pre-actions

In order to establish some of the main results of this article, we will need “cut and paste” operations on pre-actions, for instance:

* putting two prescribed pre-actions inside a single transitive action (useful for topological transitivity properties);

* modifying an action so as to add or remove a circuit in its Schreier graph (useful to get a new action that is close but distinct from the original one).

We now present these “cut and paste” operations. The main one is the following and the rest of this section will be devoted to its proof. Other useful results will appear in the course of the proof.

**Theorem 4.12 (The merging machine).** Assume $|m| \ge 2$ and $|n| \ge 2$. Let $\alpha_1$ and $\alpha_2$ two transitive non-saturated pre-actions of $\text{BS}(m, n)$ with the same phenotype. There exists a transitive action $\alpha$ which contains copies of $\alpha_1$ and $\alpha_2$ with disjoint domains.

Given a pre-action $\alpha = (\beta, \tau)$ and two sub-pre-actions $\alpha_1, \alpha_2$, let us recall that the domain of $\alpha$ is the set $\text{dom}(\beta) = \text{rng}(\beta)$. Notice that $\alpha_1$ and $\alpha_2$ have disjoint domains if and only if their Bass-Serre graphs $\text{BS}(\alpha_1)$ and $\text{BS}(\alpha_2)$ are disjoint (that is, have no common vertex) in $\text{BS}(\alpha)$.

First, taking advantage of Proposition 3.23, we reduce to the case of $(m, n)$-graphs, for which the analogous result is the following.

**Theorem 4.13 (The merging machine for $(m, n)$-graphs).** Assume $|m| \ge 2$ and $|n| \ge 2$. Let $\mathcal{G}_1$ and $\mathcal{G}_2$ be two connected and non-saturated $(m, n)$-graphs with the same phenotype. There exists a connected and saturated $(m, n)$-graph $\mathcal{G}$ which contains disjoint copies of $\mathcal{G}_1$ and $\mathcal{G}_2$.

**Remark 4.14.** The hypothesis that both $|m|, |n| \ge 2$ is necessary. If $m = 1$ but $|n| \ne 1$, we can consider the $(1, n)$-graph consisting of a single vertex with infinite label and only one loop. This graph is not saturated but it cannot be connected to another copy of itself. Indeed, the reader can check that the only saturated graph containing it admits a unique circuit, namely the loop itself.

*Proof of Theorem 4.12 based on Theorem 4.13.* The Bass-Serre graphs $\text{BS}(\alpha_1)$ and $\text{BS}(\alpha_2)$ are connected non-saturated $(m, n)$-graphs with the same phenotype. Therefore we can apply Theorem 4.13 to obtain a connected and saturated $(m, n)$-graph $\mathcal{G}$ which contains disjoint copies of $\text{BS}(\alpha_1)$ and $\text{BS}(\alpha_2)$.
• $L(\mathbf{s}(e_1)) = k$ and $L(\mathbf{t}(e_h)) = \ell$;

• the orientations of $e_1$ and $e_h$ are given by $e_1 \in E(\mathcal{G})^{\varepsilon_k}$ and $e_h \in E(\mathcal{G})^{\varepsilon_\ell}$.

*Proof.* Observe that every $(m, n)$-graph can be turned into an $(n, m)$-graph by flipping the orientations of its edges. Note that this operation does not affect the labels nor its phenotype. We thus can restrict ourselves to the case where the orientation $\varepsilon_1$ of the first edge in the path is asked to be positive and no assumption is made on $\varepsilon_h$. Let us set $q := \mathrm{Ph}_{m,n}(k) = \mathrm{Ph}_{m,n}(\ell)$.

We first treat the case $k = q = \ell$. Recall from Lemma 4.15 that $\mathrm{Ph}_{m,n}(q) = q$ and that we have $\gcd(m, q) = \gcd(n, q)$. Hence, there exists a $(m, n)$-graph with two vertices and a unique positive edge $f_1$ such that $L(\mathbf{s}(f_1)) = q = L(\mathbf{t}(f_1))$, and $L(f_1) = \frac{q}{\gcd(m,q)} = \frac{q}{\gcd(n,q)}$. If $\varepsilon_h$ is positive, we are done. If not, consider a vertex $v$ with label $L(v) = \frac{q}{\gcd(m,q)}m$. We get $\gcd(m, L(v)) = |m|$, hence $\gcd(m, L(v)) \ge 2$. Therefore, we can equip $v$ with two distinct incoming positive edges $f_1, f_2$. Such edges have to be labeled by $\frac{L(v)}{\gcd(m,L(v))} = \frac{q}{\gcd(n,q)}$ so that we can label $\mathbf{s}(f_1)$ and $\mathbf{s}(f_2)$ by $q$, and $(f_1, \bar{f}_2)$ is the path we are looking for. The theorem is thus proved for $k = \ell = q$.

Let us now treat the case $k \ne q$ and $\ell = q$. Recall that $\mathcal{P}_{m,n}(k) = \{p \in \mathcal{P} : |m|_p = |n|_p \text{ and } |n|_p < |k|_p\}$ and $\mathrm{Ph}_{m,n}(k) = \prod_{p \in \mathcal{P}_{m,n}(k)} p^{|k|_p}$. Thus any number $L \in \mathbb{Z}_{\ge 1}$ with phenotype $q$ admits a unique decomposition as follows:

$$L = q \cdot \prod_{\substack{p \in \mathcal{P} \\ |m|_p \le |n|_p}} p^{|L|_p} \prod_{\substack{p \in \mathcal{P} \\ |m|_p > |n|_p}} p^{|L|_p}. \qquad (4.18)$$

In a first step, we construct (algorithmically) a simple path consisting of positive edges with vertices $v_0, v_1, \dots, v_r$, such that $v_0$ has label $k$, and such that the decomposition of $L(v_r)$ reduces to

$$L(v_r) = q \cdot \prod_{p \in \mathcal{P} : |m|_p > |n|_p} p^{|L(v_r)|_p}, \qquad (4.19)$$

that is, such that $|L(v_r)|_p = 0$ whenever $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$.

To do so, starting with $i = 0$ and $L(v_0) = k$, while $L(v_i)$ has prime divisors $p$ such that $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$, we connect $v_i$ to a new vertex $v_{i+1}$ by a positive edge $f_i$. According to Remark 3.26, we label $f_i$ by $|L(f_i)|_p := \max(|L(v_i)|_p - |n|_p, 0)$ and set

$$|L(v_{i+1})|_p := \begin{cases} |L(f_i)|_p + |m|_p & \text{if } |L(f_i)|_p \ge 1 \\ 0 & \text{if } |L(f_i)|_p = 0 \end{cases}$$
Then, we apply Proposition 3.23 to the pre-action $\alpha_1 \sqcup \alpha_2$, whose Bass-Serre graph $\mathbf{BS}(\alpha_1) \sqcup \mathbf{BS}(\alpha_2)$ is contained in $\mathcal{G}$, to ensure the existence of a $\mathbf{BS}(m, n)$-pre-action $\alpha$ which extends $\alpha_1 \sqcup \alpha_2$. Thus $\alpha$ extends both $\alpha_1$ and $\alpha_2$ with disjoint domains. Since $\mathcal{G}$ is connected and saturated, $\alpha$ is a transitive and saturated pre-action, i.e., it is a genuine transitive action of $\mathbf{BS}(m, n)$ that satisfies the requirements of Theorem 4.12. $\square$

We now present some general results we will use in order to prove Theorem 4.13. We begin with two easy properties of phenotypes which will be useful in the proof.

**Lemma 4.15.** For any $k \in \mathbb{Z}_{\ge 1}$, if $q = \mathrm{Ph}_{m,n}(k)$, then $\mathrm{Ph}_{m,n}(q) = q$ and $\mathrm{gcd}(q, n) = \mathrm{gcd}(q, m)$.

*Proof.* We get directly from Definition 4.1 that $|q|_p = |k|_p$ if $p \in \mathcal{P}_{m,n}(k)$, and $|q|_p = 0$ for the other primes $p$. Consequently, we get $\mathcal{P}_{m,n}(q) = \mathcal{P}_{m,n}(k)$ and then $\mathrm{Ph}_{m,n}(q) = \mathrm{Ph}_{m,n}(k) = q$. Finally, since every prime $p$ dividing $q$ satisfies $|m|_p = |n|_p$ and $|n|_p < |q|_p$, we obtain

$$ \mathrm{gcd}(q, n) = \prod_{p \in \mathcal{P}: p|q} p^{|n|_p} = \prod_{p \in \mathcal{P}: p|q} p^{|m|_p} = \mathrm{gcd}(q, m). \quad \square $$

In the following lemma, by welding two vertices we mean taking the quotient graph obtained by identifying these vertices. Its proof is a direct consequence of the definition of an $(m, n)$-graph, so we omit it.

**Lemma 4.16 (Welding lemma).** Let $m, n \in \mathbb{Z} \setminus \{0\}$ and let $\mathcal{G}$ be an $(m, n)$-graph and $v, w$ be two distinct vertices such that:

* $L := L(v) = L(w)$;

* $\mathrm{deg}_{\mathrm{out}}(v) + \mathrm{deg}_{\mathrm{out}}(w) \le \mathrm{gcd}(n, L)$;

* $\mathrm{deg}_{\mathrm{in}}(v) + \mathrm{deg}_{\mathrm{in}}(w) \le \mathrm{gcd}(m, L)$.

*Welding together $v$ and $w$ delivers an $(m, n)$-graph.*

Note that in this lemma $\mathcal{G}$ can be finite or infinite, connected or not. Together with the welding lemma, the following result will allow us to connect non saturated $(m, n)$-graphs via the well-known technique of arc welding.

**Theorem 4.17 (Connecting lemma).** Assume $|m| \ge 2$ and $|n| \ge 2$. Let $k, \ell \in \mathbb{Z}_{\ge 1}$ such that $\mathrm{Ph}_{m,n}(k) = \mathrm{Ph}_{m,n}(\ell)$, and let $\varepsilon_k, \varepsilon_\ell \in \{+, -\}$. There exists an $(m, n)$-graph $\mathcal{G}$ which is a simple edge path $(e_1, \dots, e_h)$ of length $h \ge 1$ such that:
for every prime $p$. Then, we replace $i$ by $i + 1$, which terminates the “while” loop. Notice that we exit from the loop after finitely many steps. Indeed, given a prime $p$ such that $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$, we have:

* either $|L(f_1)|_p = 0$ in the case $|m|_p = |n|_p$ and $|k|_p \le |n|_p$, which implies $|L(v_i)|_p = 0$ for all $i \ge 1$;

* or $|L(v_{i+1})|_p = |L(v_i)|_p - |n|_p + |m|_p < |L(v_i)|_p$ whenever $|L(v_i)|_p \ge 1$ in the case $|m|_p < |n|_p$.

When we exit the “while” loop, Remark 3.26 guarantees that we have constructed an $(m, n)$-graph, and the loop condition guarantees that the last vertex $v_r$ satisfies $|L(v_r)|_p = 0$ whenever $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$.

If we are lucky, we have $L(v_r) = q$. If not, in a second step, we notice that the same algorithm, exchanging the roles of $m$ and $n$, produces a simple path consisting of negative edges from a vertex $w_0$ such that $L(w_0) = L(v_r)$ to a vertex $w_s$ labeled by $q$. The decomposition (4.19) of $L(v_r) \ne q$ also shows that $\gcd(m, L(v_r)) \ge 2$, so that vertices labeled $L(v_r)$ can have two distinct positive incoming edges. Using Lemma 4.16, we weld $v_r$ and $w_0$ together and get a simple path from $v_0$ to $w_s$.

In any subcase, we now have a path $(e_1, \dots, e_{h'})$ such that $e_1$ is positive, $L(\mathbf{s}(e_1)) = k$, and $L(\mathbf{t}(e_{h'})) = q$. If $e_{h'}$ has the orientation prescribed by $\varepsilon_\ell$, we are done; if not, using the case $k = q = \ell$, with the first edge having the same orientation as $e_{h'}$, and the last one having the orientation prescribed by $\varepsilon_\ell$, we extend our path to a simple path $(e_1, \dots, e_h)$ with $L(\mathbf{s}(e_1)) = k$ and $L(\mathbf{t}(e_h)) = q$ such that $e_1, e_h$ have the correct orientations. This concludes the case $\ell = q$ and $k \ne q$.

The case $k = q$ and $\ell \ne q$ is obtained by exchanging the roles of $k$ and $l$ in the above argument. Therefore, let us finally treat the case $k \ne q$ and $\ell \ne q$. The former cases furnish paths $(f_1, \dots, f_r)$ and $(f'_1, \dots, f'_s)$, that we may assume disjoint, such that

$$L(\mathbf{s}(f_1)) = k, \quad L(\mathbf{t}(f_r)) = q = L(\mathbf{s}(f'_1)), \quad L(\mathbf{t}(f'_s)) = \ell,$$

the orientations of $f_1$ and $f'_s$ are given by $\varepsilon_k$ and $\varepsilon_\ell$, and the orientations $f_r, f'_1$ coincide. Then, we just weld the vertices $\mathbf{t}(f_r)$ and $\mathbf{s}(f'_1)$ together, and the path $(f_1, \dots, f_r, f'_1, \dots, f'_s)$ is as desired. $\square$

**Remark 4.20.** In Theorem 4.17, the assumption $|m| \ge 2$ and $|n| \ge 2$ is necessary. Indeed Theorem 4.17 is false if $n = 1$. If $v$ is a vertex in a
• $L(s(e_1)) = k$ and $L(t(e_h)) = \ell$;

• the orientations of $e_1$ and $e_h$ are given by $e_1 \in E(\mathcal{G})^{\varepsilon_k}$ and $e_h \in E(\mathcal{G})^{\varepsilon_\ell}$.

*Proof.* Observe that every $(m, n)$-graph can be turned into an $(n, m)$-graph by flipping the orientations of its edges. Note that this operation does not affect the labels nor its phenotype. We thus can restrict ourselves to the case where the orientation $\varepsilon_k$ of the first edge in the path is asked to be positive and no assumption is made on $\varepsilon_\ell$. Let us set $q := \mathrm{Ph}_{m,n}(k) = \mathrm{Ph}_{m,n}(\ell)$.

We first treat the case $k = q = \ell$. Recall from Lemma 4.15 that $\mathrm{Ph}_{m,n}(q) = q$ and that we have $\gcd(m, q) = \gcd(n, q)$. Hence, there exists an $(m, n)$-graph with two vertices and a unique positive edge $f_1$ such that $L(s(f_1)) = q = L(t(f_1))$, and $L(f_1) = \frac{q}{\gcd(m,q)} = \frac{q}{\gcd(n,q)}$. If $\varepsilon_\ell$ is positive, we are done. If not, create a vertex $v$ with label $L(v) = \frac{q}{\gcd(m, L(v))}$, hence $\gcd(m, L(v)) = |m|$, hence $\gcd(m, L(v)) \ge 2$. Therefore, we can equip $v$ with two distinct incoming positive edges $f_1, f_2$. Such edges have to be labeled by $\frac{L(v)}{\gcd(m, L(v))} = \frac{q}{\gcd(n,q)}$ so that we can label $s(f_1)$ and $s(f_2)$ by $q$, and $(f_1, \bar{f}_2)$ is the path we are looking for. The theorem is thus proved for $k = \ell = q$.

Let us now treat the case $k \ne q$ and $\ell = q$. Recall that $\mathcal{P}_{m,n}(k) = \{p \in \mathcal{P} : |m|_p = |n|_p \text{ and } |n|_p < |k|_p\}$ and $\mathrm{Ph}_{m,n}(k) = \prod_{p \in \mathcal{P}_{m,n}(k)} p^{|k|_p}$. Thus any number $L \in \mathbb{Z}_{\ge 1}$ with phenotype $q$ admits a unique decomposition as follows:

$$L = q \cdot \prod_{\substack{p \in \mathcal{P} \\ |m|_p \le |n|_p}} p^{|L|_p} \prod_{\substack{p \in \mathcal{P} \\ |m|_p > |n|_p}} p^{|L|_p} \quad (4.18)$$

In a first step, we construct (algorithmically) a simple path consisting of positive edges with vertices $v_0, v_1, \dots, v_r$, such that $v_0$ has label $k$, and such that the decomposition of $L(v_r)$ reduces to

$$L(v_r) = q \cdot \prod_{p \in \mathcal{P} : |m|_p > |n|_p} p^{|L(v_r)|_p}, \quad (4.19)$$

that is, such that $|L(v_r)|_p = 0$ whenever $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$.

To do so, starting with $i = 0$ and $L(v_0) = k$, while $L(v_i)$ has prime divisors $p$ such that $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$, we connect $v_i$ to a new vertex $v_{i+1}$ by a positive edge $f_i$. According to Remark 3.26, we label $f_i$ by $|L(f_i)|_p := \max(|L(v_i)|_p - |n|_p, 0)$ and set

$$|L(v_{i+1})|_p := \begin{cases} |L(f_i)|_p + |m|_p & \text{if } |L(f_i)|_p \ge 1 \\ 0 & \text{if } |L(f_i)|_p = 0 \end{cases}$$
(m, 1)-graph with $L(v) = 1$ and $e$ is an edge such that $\mathbf{t}(e) = v$, then

$$1 = L(\mathbf{t}(e)) = \frac{L(\mathbf{t}(e))}{\gcd(L(\mathbf{t}(e)), m)} = \frac{L(\mathbf{s}(e))}{\gcd(L(\mathbf{s}(e)), 1)} = L(\mathbf{s}(e)).$$

Clearly any vertex with label 1 has at most one outgoing and one incoming edge. This implies that the labels of the vertices in any directed path which end in $v$ must be all 1. In other words, if we have any simple edge path as in Theorem 4.17 such that $\ell = 1$ and $\varepsilon_{\ell} = -$, then we must have that $k = 1$ (and $\varepsilon_k = +$).

**Lemma 4.21 (Forest-saturation lemma).** Let $\mathcal{G}$ be a connected $(m, n)$-graph. There is a saturated and connected $(m, n)$-graph $\mathcal{G}'$ containing $\mathcal{G}$ and such that:

* the subgraph induced in $\mathcal{G}'$ by $V(\mathcal{G})$ is exactly $\mathcal{G}$;

* the subgraph induced in $\mathcal{G}'$ by $V(\mathcal{G}') \setminus V(\mathcal{G})$ is a forest $\mathcal{F}$;

* all vertices of $\mathcal{F}$ have degree $\ge 1 + \min(|m|, |n|)$ in $\mathcal{G}'$;

* each connected component of $\mathcal{F}$ is connected to $\mathcal{G}$ by a single edge of $\mathcal{G}'$.

**Definition 4.22.** We call **forest-saturation** of $\mathcal{G}$ any extension $\mathcal{G}'$ satisfying Lemma 4.21. The graph $\mathcal{G}'$ produced in this proof will be called the **maximal forest-saturation** of $\mathcal{G}$.

*Proof of Lemma 4.21.* We can assume that the connected graph $\mathcal{G}$ is not yet saturated: it admits non-saturated vertices i.e., vertices $v$ for which one of the inequalities (3.14) $\deg_{\text{out}}(v) \le \gcd(L(v), n)$ or $\deg_{\text{in}}(v) \le \gcd(L(v), m)$ is strict. For every non-saturated vertex $v$ of $\mathcal{G}$ we add

* $(\gcd(L(v), n) - \deg_{\text{out}}(v))$-many new $v$-outgoing edges labeled $L_{\text{out}} := \frac{L(v)}{\gcd(n, L(v))}$ with extra target vertices labeled $mL_{\text{out}}$; and

* $(\gcd(L(v), m) - \deg_{\text{in}}(v))$-many new $v$-incoming edges labeled $L_{\text{in}} := \frac{L(v)}{\gcd(m, L(v))}$ with extra source vertices labeled $nL_{\text{in}}$.

We then iterate this construction. All the non-saturated vertices of the $j$-th step become saturated at the $(j+1)$-th one. The increasing union $\mathcal{G}'$ of these $(m, n)$-graphs is a saturated $(m, n)$-graph. The complement of $\mathcal{G}$ in it is a forest since at each step, each new edge has a new vertex as one of its vertices. The label of each new vertex $v$ is an integer multiple of either $m$ or $n$. Thus the degree $\deg_{\text{out}}(v) + \deg_{\text{in}}(v) = \gcd(L(v), n) + \gcd(L(v), m)$ of $v$ is larger than $1 + \min(|m|, |n|)$ as expected. $\square$
for every prime $p$. Then, we replace $i$ by $i + 1$, which terminates the “while” loop. Notice that we exit from the loop after finitely many steps. Indeed, given a prime $p$ such that $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$, we have:

* either $|L(f_1)|_p = 0$ in the case $|m|_p = |n|_p$ and $|k|_p \le |n|_p$, which implies $|L(v_i)|_p = 0$ for all $i \ge 1$;

* or $|L(v_{i+1})|_p = |L(v_i)|_p - |n|_p + |m|_p < |L(v_i)|_p$ whenever $|L(v_i)|_p \ge 1$ in the case $|m|_p < |n|_p$.

When we exit the “while” loop, Remark 3.26 guarantees that we have constructed an $(m, n)$-graph, and the loop condition guarantees that the last vertex $v_r$ satisfies $|L(v_r)|_p = 0$ whenever $|m|_p \le |n|_p$ and $p \notin \mathcal{P}_{m,n}(k)$.

If we are lucky, we have $L(v_r) = q$. If not, in a second step, we notice that the same algorithm, exchanging the roles of $m$ and $n$, produces a simple path consisting of negative edges from a vertex $w_0$ such that $L(w_0) = L(v_r)$ to a vertex $w_s$ labeled by $q$. The decomposition (4.19) of $L(v_r) \ne q$ also shows that $\gcd(m, L(v_r)) \ge 2$, so vertices labeled $L(v_r)$ can have two distinct positive incoming edges. Using Lemma 4.16, we weld $v_r$ and $w_0$ together and get a simple path from $v_0$ to $w_s$.

In any subcase, we now have a path $(e_1, \dots, e_{h'})$ such that $e_1$ is positive, $L(\mathbf{s}(e_1)) = k$, and $L(\mathbf{t}(e_{h'})) = q$. If $e_{h'}$ has the orientation prescribed by $\varepsilon_\ell$, we are done; if not, using the case $k = q = \ell$, with the first edge having the same orientation as $e_{h'}$, and the last one having the orientation prescribed by $\varepsilon_\ell$, we extend our path to a simple path $(e_1, \dots, e_h)$ with $L(\mathbf{s}(e_1)) = k$ and $L(\mathbf{t}(e_h)) = q$ such that $e_1, e_h$ have the correct orientations. This concludes the case $\ell = q$ and $k \ne q$.

The case $k = q$ and $\ell \ne q$ is obtained by exchanging the roles of $k$ and $l$ in the above argument. Therefore, let us finally treat the case $k \ne q$ and $\ell \ne q$. The former cases furnish paths $(f_1, \dots, f_r)$ and $(f'_1, \dots, f'_s)$, that we may assume disjoint, such that

$$L(\mathbf{s}(f_1)) = k, \quad L(\mathbf{t}(f_r)) = q = L(\mathbf{s}(f'_1)), \quad L(\mathbf{t}(f'_s)) = \ell,$$

the orientations of $f_1$ and $f'_s$ are given by $\varepsilon_k$ and $\varepsilon_\ell$, and the orientations $f_r, f'_1$ coincide. Then, we just weld the vertices $\mathbf{t}(f_r)$ and $\mathbf{s}(f'_1)$ together, and the path $(f_1, \dots, f_r, f'_1, \dots, f'_s)$ is as desired. $\square$

**Remark 4.20.** In Theorem 4.17, the assumption $|m| \ge 2$ and $|n| \ge 2$ is necessary. Indeed Theorem 4.17 would be false for $n = 1$. If $v$ is a vertex in
While the labels of the new edges are prescribed by the axiomatic of $(m, n)$-graphs, we made the choice of the maximal label for the new vertices among those satisfying the equation (3.13) $\frac{L(s(e))}{\gcd(L(s(e)),n)} = L(e) = \frac{L(t(e))}{\gcd(L(t(e)),m)}$. Hence the terminology in Definition 4.22.

*Proof of Theorem 4.13.* By hypothesis, for $i = 1, 2$, there is a non-saturated vertex $v_i$ in $G_i$, i.e., a vertex that misses an edge with terminal vertex $v_i$ and orientation $\epsilon_i \in \{+, -\}$. The labels of $v_1, v_2$ having identical phenotypes, the connecting Theorem 4.17 furnishes an $(m, n)$-graph $G_0$ which is a simple edge path $(e_1, \dots, e_h)$ such that $L(s(e_1)) = L(v_1)$ and $L(t(e_h)) = L(v_2)$, and the orientations of $e_1$ and $e_h$ are given by $-\epsilon_1$ and $\epsilon_2$ respectively.

We then consider the disjoint union $G_1 \sqcup G_0 \sqcup G_2$. We claim that we can merge the vertices $v_1$ and $s(e_1)$ thanks to the welding Lemma 4.16. Indeed, the choice of orientation for $e_1$ and the form of $G_0$ (a path of edges) are made for the assumptions of Lemma 4.16 to hold. Then, we can merge $v_2$ and $t(e_h)$, applying Lemma 4.16 again (this time, using the fact that the orientation of $e_h$ is well chosen). This produces a connected $(m, n)$-graph $G_3$ which contains disjoint copies of $G_1$ and $G_2$.

It now suffices to apply the saturation Lemma 4.21 to $G_3$ so as to obtain a connected saturated $(m, n)$-graph $G$ that satisfies the requirements of Theorem 4.13. □

# 5 Perfect kernel and dense orbits

## 5.1 Perfect kernels of Baumslag-Solitar groups

In case $|m| = 1$ or $|n| = 1$, it follows from the proof of [BLT19, Cor. 8.4] that Sub($BS(m, n)$) is countable, hence the perfect kernel $K(\mathrm{BS}(m, n))$ is empty. Our main theorem describes the perfect kernels in the remaining cases.

**Theorem 5.1.** Let $m, n \in \mathbb{Z}$ such that $|m| \ge 2$ and $|n| \ge 2$. We have

$$ \mathcal{K}(\mathrm{BS}(m, n)) = \{ \Lambda \in \mathrm{Sub}(\mathrm{BS}(m, n)) : \Lambda \setminus \mathrm{BS}(m, n) / \langle b \rangle \text{ is infinite} \}. $$

Let us temporarily give a name to the set appearing in Theorem 5.1:

$$ \mathcal{L} = \mathcal{L}(m, n) := \{ \Lambda \in \mathrm{Sub}(\mathrm{BS}(m, n)) : \Lambda \setminus \mathrm{BS}(m, n) / \langle b \rangle \text{ is infinite} \}, $$

and recall that $\mathrm{Sub}_{[\infty]}(\Gamma)$ denotes the space of infinite index subgroups of $\Gamma$.
a $(m, 1)$-graph with $L(v) = 1$ and $e$ is an edge such that $\mathfrak{t}(e) = v$, then

$$1 = L(\mathfrak{t}(e)) = \frac{L(\mathfrak{s}(e))}{\gcd(L(\mathfrak{t}(e)), m)} = \frac{L(\mathfrak{s}(e))}{\gcd(L(\mathfrak{s}(e)), 1)} = L(\mathfrak{s}(e)).$$

Clearly any vertex with label 1 has at most one outgoing and one incoming edge. This implies that the labels of the vertices in any directed path which ends in $v$ must be all 1. In other words, if we have any simple edge path as in Theorem 4.17 such that $\ell = 1$ and $\varepsilon_{\ell} = -$, then we must have that $k = 1$ (and $\varepsilon_k = +$).

**Definition 4.21.** Let $\mathcal{G}$ be a connected $(m, n)$-graph. A saturated extension $\mathcal{G}'$ of $\mathcal{G}$ is called a **forest-saturation** of $\mathcal{G}$ if it satisfies

* the subgraph induced in $\mathcal{G}'$ by $V(\mathcal{G})$ is exactly $\mathcal{G}$;

* the subgraph induced in $\mathcal{G}'$ by $V(\mathcal{G}') \setminus V(\mathcal{G})$ is a forest $\mathcal{F}$;

* each connected component of $\mathcal{F}$ is connected to $\mathcal{G}$ by a single edge of $\mathcal{G}'$.

**Lemma 4.22 (Forest-saturation lemma).** Let $\mathcal{G}$ be a connected $(m, n)$-graph. There is a forest-saturation $\mathcal{G}'$ of $\mathcal{G}$ such that all vertices of the forest $\mathcal{F}$ induced in $\mathcal{G}'$ by $V(\mathcal{G}') \setminus V(\mathcal{G})$ have degree $\ge 1 + \min(|m|, |n|)$ in $\mathcal{G}'$.

The reader can observe in the following construction proving Lemma 4.22 that, while the labels of the new edges are prescribed, the axioms of $(m, n)$-graphs allows some choices concerning the labels of the new vertices. The systematic choice of the maximal label will be made for the new vertices among all those satisfying the transfer equation (3.13) $\frac{L(\mathfrak{s}(e))}{\gcd(L(\mathfrak{s}(e)), n)} = L(e) = \frac{L(\mathfrak{t}(e))}{\gcd(L(\mathfrak{t}(e)), m)}$. Hence the forest-saturation constructed in this proof is called the **maximal forest-saturation** of $\mathcal{G}$. Notice that other choices would have led to forest-saturations with different underlying graphs, by virtue of the relationship between labels and degrees (see Definition 3.20). These forest-saturations are further studied in the recent preprint [GLMS24].

*Proof of Lemma 4.22.* We can assume that the connected graph $\mathcal{G}$ is not yet saturated: it admits non-saturated vertices i.e., vertices $v$ for which one of the inequalities (3.14) $\deg_{\text{out}}(v) \le \gcd(L(v), n)$ or $\deg_{\text{in}}(v) \le \gcd(L(v), m)$ is strict. For every non-saturated vertex $v$ of $\mathcal{G}$ we add

* $(\gcd(L(v), n) - \deg_{\text{out}}(v))$-many new $v$-outgoing edges labeled $L_{\text{out}} := \frac{L(v)}{\gcd(n, L(v))}$ with extra target vertices labeled $mL_{\text{out}}$; and
Given an action $\alpha$ of $\Gamma$ on a space $X$ and a point $v \in X$, we have already introduced the notation $[\alpha, v]$ for the action $\alpha$ pointed at $v$.

**Remark 5.2.** In terms of pointed transitive actions, $\mathcal{L}(m, n)$ is the set of pointed transitive actions with infinitely many $b$-orbits, whence $\mathcal{L} = \{[\alpha, v]: \mathbf{BS}(\alpha)$ is infinite\}$. Moreover:

* if $|m| \neq |n|$, we have $\mathcal{L}(m, n) = \mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n))$, since every infinite action has an infinite Bass-Serre graph by Lemma 4.11.

* if $|m| = |n|$, we have $\mathcal{L}(m, n) = \pi^{-1}(\mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n)/\langle b^m \rangle))$, where $\pi$ is the homomorphism from $\mathrm{BS}(m, n)$ to its quotient by the normal subgroup $\langle b^m \rangle = \langle b^n \rangle$

$$1 \to \langle b^m \rangle \to \mathrm{BS}(m, n) \xrightarrow{\pi} \mathrm{BS}(m, n)/\langle b^m \rangle \to 1.$$

Indeed, since $\langle b^m \rangle$ has finite index in $\langle b \rangle$, we get that $\Lambda\backslash\mathrm{BS}(m, n)/\langle b \rangle$ is finite if and only if $\Lambda\backslash\mathrm{BS}(m, n)/\langle b^m \rangle$ is finite.

Therefore, Theorem 5.1 can be rephrased in two ways, as follows.

**Theorem 5.3.** Let $m, n \in \mathbb{Z}$ such that $|m| \geq 2$ and $|n| \geq 2$.

1. In terms of pointed transitive actions, the perfect kernel corresponds exactly to actions whose Bass-Serre graph is infinite:

$$\mathcal{K}(\mathrm{BS}(m, n)) = \{[\alpha, v]: \mathbf{BS}(\alpha) \text{ is infinite}\}.$$

2. In terms of subgroups:

* if $|m| \neq |n|$, the perfect kernel is equal to the space of infinite index subgroups:

$$\mathcal{K}(\mathrm{BS}(m, n)) = \mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n));$$

* if $|m| = |n|$, we have:

$$\mathcal{K}(\mathrm{BS}(m, n)) = \pi^{-1}(\mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n)/\langle b^m \rangle)),$$

where $\pi$ is the homomorphism from $\mathrm{BS}(m, n)$ to its quotient by the normal subgroup $\langle b^m \rangle = \langle b^n \rangle$. $\square$

*Proof of Theorem 5.1.* Our aim is to prove that $\mathcal{K}(\mathrm{BS}(m, n)) = \mathcal{L}(m, n)$. It will be convenient to write one inclusion in terms of pointed transitive actions and the other in terms of subgroups.

Let us first prove the inclusion $\mathcal{K}(\mathrm{BS}(m, n)) \supseteq \mathcal{L}$. It suffices to show that no element of $\mathcal{L}$ is isolated in $\mathcal{L}$. Recall the definition of the topology in
• $(\gcd(L(v), m) - \deg_{\text{in}}(v))$-many new $v$-incoming edges labeled $L_{\text{in}} := \frac{L(v)}{\gcd(m, L(v))}$ with extra source vertices labeled $nL_{\text{in}}$.

We then iterate this construction. All the non-saturated vertices of the $j$-th step become saturated at the $(j+1)$-th one. The increasing union $\mathcal{G}'$ of these $(m, n)$-graphs is a saturated $(m, n)$-graph. The complement of $\mathcal{G}$ in it is a forest since at each step, each new edge has a new vertex as one of its vertices. The label of each new vertex $v$ is an integer multiple of either $m$ or $n$. Thus the degree $\deg_{\text{out}}(v) + \deg_{\text{in}}(v) = \gcd(L(v), n) + \gcd(L(v), m)$ of $v$ is larger than $1 + \min(|m|, |n|)$ as expected. $\square$

*Proof of Theorem 4.13.* By hypothesis, for $i = 1, 2$, there is a non-saturated vertex $v_i$ in $\mathcal{G}_i$; i.e. a vertex for which one of the inequalities (3.14) is strict. If $\deg_{\text{in}}(v_i) < \gcd(L(v_i), m)$, then let $\epsilon_i := +$; otherwise let $\epsilon_i := -$. The labels of $v_1, v_2$ having identical phenotypes, the connecting lemma (Theorem 4.17) furnishes an $(m, n)$-graph $\mathcal{G}_0$ which is a simple edge path $(e_1, \dots, e_h)$ such that $L(\mathfrak{s}(e_1)) = L(v_1)$ and $L(\mathfrak{t}(e_h)) = L(v_2)$, and the orientations of $e_1$ and $e_h$ are given by $-\epsilon_1$ and $\epsilon_2$ respectively.

We then consider the disjoint union $\mathcal{G}_1 \sqcup \mathcal{G}_0 \sqcup \mathcal{G}_2$. We claim that we can merge the vertices $v_1$ and $\mathfrak{s}(e_1)$ thanks to the welding Lemma 4.16. Indeed, the choice of orientation for $e_1$ and the form of $\mathcal{G}_0$ (a path of edges) are made for the assumptions of Lemma 4.16 to hold. Then, we can merge $v_2$ and $\mathfrak{t}(e_h)$, applying Lemma 4.16 again (this time, using the fact that the orientation of $e_h$ is well chosen). This produces a connected $(m, n)$-graph $\mathcal{G}_3$ which contains disjoint copies of $\mathcal{G}_1$ and $\mathcal{G}_2$.

It now suffices to apply the saturation Lemma 4.22 to $\mathcal{G}_3$ so as to obtain a connected saturated $(m, n)$-graph $\mathcal{G}$ that satisfies the requirements of Theorem 4.13. $\square$

# 5 Perfect kernel and dense orbits

## 5.1 Perfect kernels of Baumslag-Solitar groups

In case $|m| = 1$ or $|n| = 1$, it follows from the proof of [BLT19, Cor. 8.4] that $\text{Sub}(\text{BS}(m, n))$ is countable, hence the perfect kernel $\mathcal{K}(\text{BS}(m, n))$ is empty. Our main theorem describes the perfect kernels in the remaining cases.

**Theorem 5.1.** Let $m, n \in \mathbb{Z}$ such that $|m| \ge 2$ and $|n| \ge 2$. We have

$$ \mathcal{K}(\text{BS}(m, n)) = \{ \Lambda \in \text{Sub}(\text{BS}(m, n)) : \Lambda \setminus \text{BS}(m, n) / \langle b \rangle \text{ is infinite} \}. $$
terms of pointed actions, see Section 2.2 and in particular Equation (2.7). Let us fix a pointed transitive action $(\alpha_0, v)$ representing an element of $\mathcal{L}$ and a radius $R \ge 0$. We will show that the basic neighborhood $\mathcal{N}([\alpha_0, v], R)$ contains at least two distinct elements of $\mathcal{L}$.

Let $(\beta, \tau)$ be the pre-action obtained by restricting $\alpha_0$ to the reunion of the $b$-orbits of the vertices of the ball of radius $R+1$ centered at $v$ in the Schreier graph of $\alpha_0$. The Bass-Serre graph of $(\beta, \tau)$ is the projection in $\mathbf{BS}(\alpha_0)$ (see Definition 3.10) of this ball, hence is finite. Since $\mathbf{BS}(\alpha_0)$ is infinite, the pre-action $(\beta, \tau)$ is not saturated.

We now build two $(m, n)$-graphs $G_1, G_2$ that extend the finite non-saturated Bass-Serre graph $\mathcal{G}$ of $(\beta, \tau)$ in two different ways. First, let $G_1$ be a forest-saturation of $\mathcal{G}$ see (Definition 4.22). Let us recall that the subgraph induced in $G_1$ by $V(G_1)\setminus V(\mathcal{G})$ is a forest whose vertices have degree at least three in $G_1$.

We then construct $G_2$ by modifying $G_1$. Let us pick a vertex $v \in V(G_1)\setminus V(\mathcal{G})$. The subgraph induced in $G_1$ by $V(G_1)\setminus\{v\}$ has at least three connected components. Choose two connected components disjoint from $\mathcal{G}$ and remove them. In the resulting $(m, n)$-graph $G_1'$, the vertex $v$ is the only one that is not saturated: two edges are missing.

Theorem 4.17 gives us an $(m, n)$-graph which is a simple edge path $\mathcal{P}$ whose extremities have the same label as $v$ and for which the orientations of the end edges are compatible with that of the missing edges of $v$. We then apply twice the welding lemma, Lemma 4.16, so as to weld the two extremities of $\mathcal{P}$ to $v$. We eventually define $G_2$ to be a forest-saturation of the graph that we obtained. Observe that $G_1$ is not isomorphic to $G_2$ since the fundamental groups of their underlying graphs are free groups of distinct ranks.

Finally, we extend $(\beta, \tau)$ to pre-actions $\alpha_1$ and $\alpha_2$ whose Bass-Serre graphs are $G_1$ and $G_2$ respectively, thanks to Proposition 3.23. Since $G_1, G_2$ are saturated, $\alpha_1, \alpha_2$ are actually actions by Example 3.21. We already remarked that $G_1$ is not isomorphic to $G_2$, so the pointed transitive actions $(\alpha_1, v)$ and $(\alpha_2, v)$ are not isomorphic: $[\alpha_1, v] \ne [\alpha_2, v]$. Moreover, the balls of radius $R$ centered at the basepoints in the Schreier graphs of $\alpha_0, \alpha_1, \alpha_2$ all coincide by construction with that of $(\beta, \tau)$, so $[\alpha_1, v]$ and $[\alpha_2, v]$ are both in $\mathcal{N}([\alpha_0, v], R)$.

Let us now turn to the inclusion $\mathcal{K}(\mathbf{BS}(m, n)) \subseteq \mathcal{L}$. Let us pick a subgroup $\Lambda \in \text{Sub}(\mathbf{BS}(m, n))\setminus\mathcal{L}(m, n)$ and let us prove that it is not in the perfect kernel.
Let us temporarily give a name to the set appearing in Theorem 5.1:

$$ \mathcal{J} = \mathcal{J}(m, n) := \{ \Lambda \in \mathrm{Sub}(\mathrm{BS}(m, n)) : \Lambda \setminus \mathrm{BS}(m, n) / \langle b \rangle \text{ is infinite} \}, $$

and recall that $\mathrm{Sub}_{[\infty]}(\Gamma)$ denotes the space of infinite index subgroups of $\Gamma$.

Given an action $\alpha$ of $\Gamma$ on a space $X$ and a point $v \in X$, we have already introduced the notation $[\alpha, v]$ for the action $\alpha$ pointed at $v$.

**Remark 5.2.** In terms of pointed transitive actions, $\mathcal{J}(m, n)$ is the set of pointed transitive actions with infinitely many $b$-orbits, whence $\mathcal{J} = \{[\alpha, v]: \mathbf{BS}(\alpha)$ is infinite\}$. Moreover:

* if $|m| \neq |n|$, we have $\mathcal{J}(m, n) = \mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n))$, since every infinite action has an infinite Bass-Serre graph by Lemma 4.11.

* if $|m| = |n|$, we have $\mathcal{J}(m, n) = \pi^{-1}(\mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n)/\langle b^m \rangle))$, where $\pi$ is the homomorphism from $\mathrm{BS}(m, n)$ to its quotient by the normal subgroup $\langle b^m \rangle = \langle b^n \rangle$. Indeed, since $\langle b^m \rangle$ has finite index in $\langle b \rangle$, we get that $\Lambda \setminus \mathrm{BS}(m, n)/\langle b \rangle$ is finite if and only if $\Lambda \setminus \mathrm{BS}(m, n)/\langle b^m \rangle$ is finite.

Therefore, Theorem 5.1 can be rephrased in two ways, as follows.

**Theorem 5.3.** Let $m, n \in \mathbb{Z}$ such that $|m| \ge 2$ and $|n| \ge 2$.

1. In terms of pointed transitive actions, the perfect kernel corresponds exactly to actions whose Bass-Serre graph is infinite:

$$ \mathcal{K}(\mathrm{BS}(m, n)) = \{[\alpha, v]: \mathbf{BS}(\alpha) \text{ is infinite}\}. $$

2. In terms of subgroups:

* if $|m| \neq |n|$, the perfect kernel is equal to the space of infinite index subgroups:

$$ \mathcal{K}(\mathrm{BS}(m, n)) = \mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n)); $$

* if $|m| = |n|$, we have:

$$ \mathcal{K}(\mathrm{BS}(m, n)) = \pi^{-1}(\mathrm{Sub}_{[\infty]}(\mathrm{BS}(m, n)/\langle b^m \rangle)), $$

where $\pi$ is the homomorphism from $\mathrm{BS}(m, n)$ to its quotient by the normal subgroup $\langle b^m \rangle = \langle b^n \rangle$. $\square$

*Proof of Theorem 5.1.* Our aim is to prove that $\mathcal{K}(\mathrm{BS}(m, n)) = \mathcal{J}(m, n)$. It will be convenient to write one inclusion in terms of pointed transitive actions and the other in terms of subgroups.
If $|m| \neq |n|$, then $\Lambda$ has finite index in $\text{BS}(m, n)$ by Remark 5.2, hence it is isolated in $\text{Sub}(\text{BS}(m, n))$.

If $|m| = |n|$, then $\pi(\Lambda)$ has finite index in $\text{BS}(m, n)/\langle b^m \rangle$ by Remark 5.2, hence it is finitely generated. Therefore, the set

$$ \mathcal{V} := \{ \Lambda' \in \text{Sub}(\text{BS}(m, n)) : \pi(\Lambda') \geq \pi(\Lambda) \} $$

is a neighborhood of $\Lambda$, since it contains the basic neighborhood $\mathcal{V}(S, \emptyset) = \{\Lambda' \in \text{Sub}(\text{BS}(m, n)): S \subseteq \Lambda'\}$ where $S \subseteq \Lambda$ is a finite set such that $\pi(S)$ generates $\pi(\Lambda)$.

Now, for any $\Lambda' \in \mathcal{V}$, the subgroup $\pi(\Lambda')$ has finite index in $\text{BS}(m, m)/\langle b^m \rangle$. Hence $\pi(\Lambda')$ is finitely generated, so that $\Lambda'$ itself is finitely generated as it is written as an extension with cyclic kernel:

$$ 1 \to \langle b^m \rangle \cap \Lambda' \to \Lambda' \to \pi(\Lambda') \to 1. $$

Therefore all subgroups of $\mathcal{V}$ are finitely generated, which implies that $\mathcal{V}$ is countable and hence $\Lambda$ is not in $\mathcal{K}(\text{BS}(m, n))$. $\square$

**Corollary 5.4.** If $|m| \geq 2$, $|n| \geq 2$ and $|m| \neq |n|$, then

$$ \mathbf{Ph}^{-1}(\infty) \subseteq \mathcal{K}(\text{BS}(m, n)); $$

in other words, every infinite phenotype subgroup is in the perfect kernel.

*Proof.* Any subgroup with infinite phenotype has infinite index and hence it is in $\mathcal{K}(\text{BS}(m, n))$ according to Theorem 5.3. $\square$

## 5.2 Phenotypical decomposition of the perfect kernel

Let us now turn to a description of the internal structure of $\mathcal{K}(\text{BS}(m, n))$.

**Notation 5.5.** Let $m, n \in \mathbb{Z}\{-1, 0, 1\}$. We denote by $\mathcal{Q}_{m,n}$ the set of all possible $(m, n)$-phenotypes, that is, $\mathcal{Q}_{m,n} := \text{Ph}_{m,n}(\mathbb{Z}_{\geq 1} \cup \{\infty\})$.

**Definition 5.6.** The phenotype of a subgroup $\Lambda \leq \text{BS}(m, n)$ is the phenotype of the associated action defined in Definition 4.9:

$$ \mathbf{Ph}(\Lambda) := \mathbf{Ph}(\Lambda \setminus \text{BS}(m, n) \hookrightarrow \text{BS}(m, n)). $$

This yields a function $\mathbf{Ph}: \text{Sub}(\text{BS}(m, n)) \to \mathcal{Q}_{m,n} \subseteq \mathbb{Z}_{\geq 1} \cup \{\infty\}$.