Chabauty limits of simple groups acting on trees
Pierre-Emmanuel Caprace∗1 and Nicolas Radu†1

arXiv:1608.00461v2 [] 25 Nov 2016

1

UCLouvain, 1348 Louvain-la-Neuve, Belgium

November 25, 2016

Abstract
Let T be a locally finite tree without vertices of degree 1. We show that among
closed subgroups of Aut(T ) acting with a bounded number of orbits, the Chabautyclosure of the set of topologically simple groups is the set of groups without proper
open subgroup of finite index. Moreover, if all vertices of T have degree ≥ 3, then
the set of isomorphism classes of topologically simple closed subgroups of Aut(T ) acting doubly transitively on ∂T carries a natural compact Hausdorff topology inherited
from Chabauty. Some of our considerations are valid in the context of automorphism
groups of locally finite connected graphs. Applications to Weyl-transitive automorphism groups of buildings are also presented.

Contents
1 Introduction

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2 The Chabauty space

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3 The k-closure of a graph automorphism group

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4 Finite quotients of groups acting on graphs
5 Trees
5.1 Existence and conjugation of tree lattices . . . . . .
5.2 Limits of simple groups acting on trees . . . . . . . .
5.3 Boundary-2-transitive automorphism groups of trees
5.4 Local prime content and local torsion-freeness . . . .

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6 Buildings
22
6.1 Weyl-transitive automorphism groups of buildings . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Buildings of virtually free type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
A The clopen subset STAlt ⊆ ST
∗
†

24

F.R.S.-FNRS Senior Research Associate, supported in part by the ERC (grant #278469).
F.R.S.-FNRS Research Fellow.

1

1

Introduction
Hêtre c’est mon identité
(Jacques Prévert, Arbres, 1976)

Beyond algebraic groups over local fields, groups acting on trees provide the largest
(and historically the first) known source of examples of non-discrete compactly generated
locally compact groups that are topologically simple, i.e. whose only closed normal
subgroups are the trivial ones. Since the automorphism group of a given locally finite
tree T may host many pairwise non-isomorphic topologically simple closed subgroups,
it is natural to consider those collectively, by viewing them as a subset of the space
Sub(Aut(T )) of all closed subgroups of Aut(T ), endowed with the Chabauty topology,
which is compact. The starting point of this work is the following basic question: what
is the Chabauty-closure of the set of topologically simple closed subgroups of Aut(T )? In
order to stay in the realm of compactly generated groups, we will frequently impose that
the groups under consideration act with a bounded number of orbits. Assuming the
weaker condition that the groups act cocompactly on T is sufficient to guarantee that
they are compactly generated, but that condition is not Chabauty-closed. To facilitate
the statements of our results, we introduce the following notation. For a given number
C > 0, we denote by
Sub(Aut(T ))≤C
the set of closed subgroups of Aut(T ) acting with at most C orbits of vertices. It is a
clopen subset of Sub(Aut(T )) (see Proposition 2.6 (3)).
Theorem 1.1. Let T be a locally finite tree all of whose vertices have degree ≥ 2. For any
C > 0, the Chabauty-closure of the set of topologically simple groups in Sub(Aut(T ))≤C
is the set of groups in Sub(Aut(T ))≤C without proper open subgroup of finite index.
The conclusion of Theorem 1.1 may fail if the tree T is allowed to have vertices of
degree 1, see Lemma 5.14 below.
Following Burger–Mozes [BM00], it is customary to denote the intersection of all open
subgroups of finite index in a given locally compact group H by H (∞) . We also denote by
Mon(H) the monolith of H, i.e. the (possibly trivial) intersection of all non-trivial closed
normal subgroups of H. Notice that H is topologically simple if and only if H = Mon(H).
With these notations at hand, the statement of Theorem 1.1 can be epitomized by the
following equality:
{H ∈ Sub(Aut(T ))≤C | H = Mon(H)} = {H ∈ Sub(Aut(T ))≤C | H = H (∞) }.
We remark that if C = 1 then the set {H ∈ Sub(Aut(T ))≤C | H = H (∞) } is empty,
while if C ≥ 2 and T is semi-regular (i.e. Aut(T ) is edge-transitive), that set contains at
least one group, namely the group Aut(T )+ of type-preserving automorphisms, which is
simple by [Tit70]. For a general tree T and an arbitrarily large C, it may be the case that
Sub(Aut(T ))≤C contains only discrete, hence virtually free, groups (see [BT]), so that the
set {H ∈ Sub(Aut(T ))≤C | H = H (∞) } is also empty in that case.
2

It is important to note that a Chabauty limit of topologically simple groups need not
be simple. Indeed, explicit examples of non-simple closed subgroups H of Aut(T ) that
are edge-transitive (indeed locally 2-transitive) and satisfy H = H (∞) are provided by
Burger and Mozes in [BM00, Example 1.2.1] (see also Remark 5.13 below). Thus the set
of topologically simple cocompact closed subgroups of bounded covolume is not closed in
Sub(Aut(T )). Nevertheless, that situation changes if one considers the subset of groups
acting doubly transitively on the set of ends of a thick tree T (which is automatically
contained in Sub(Aut(T ))≤2 ). Recall that T is thick if all its vertices have degree ≥ 3.
Theorem 1.2. Let T be a locally finite thick tree. The set of topologically simple closed
subgroups of Aut(T ) acting 2-transitively on ∂T is Chabauty-closed.
Moreover, the isomorphism relation within that set has closed classes. Hence, the set
ST of isomorphism classes of topologically simple groups acting continuously and properly
on T and 2-transitively on ∂T , endowed with the quotient topology, is compact Hausdorff.
Theorem 1.2 has several consequences. First of all, it can be interpreted as providing
qualitative information on the complexity of the isomorphism relation within topologically
simple boundary-2-transitive closed subgroups of Aut(T ). Indeed, Theorem 1.2 implies
that that relation is smooth in the sense of [Gao09, Definition 5.4.1], which means that
it comes at the bottom of the hierarchy of complexity of classification problems in the
formalism established by invariant descriptive set theory (see [Gao09, Chapter 15]). In
fact, it is tantalizing to believe that for a given tree T , the set ST of isomorphism classes as
above can be described exhaustively. This has actually recently been accomplished by the
second-named author for all trees whose vertex degrees are ≥ 6 and such that the only finite
2-transitive groups of that degree are the full symmetric or alternating groups, see [Rad15]
and Appendix A below. For all those trees, the set ST happens to be countable. Moreover,
as a compact totally disconnected space, the second Cantor–Bendixson derivative of ST
is reduced to the singleton consisting of the isomorphism class of the group Aut(T )+ (see
Proposition A.1 and Remark A.3 below). However, the classification problem remains
open for semi-regular trees T whose vertex degrees are the degrees of smaller finite 2transitive groups, like Lie-type groups or affine groups. In particular, we do not know
whether there exists a tree T such that ST is uncountable. The case of the trivalent tree
is especially intriguing.
The compactness of ST asserted by Theorem 1.2 also fosters less ambitious hope than
a full classification of ST . Indeed, it opens up the possibility to find new isomorphism
types of simple groups by taking limits of known ones. Implementing this idea requires
to have at hand an infinite family of pairwise non-isomorphic topologically simple groups
acting boundary-2-transitively on the same locally finite tree T . Rank one simple algebraic
groups over p-adic fields provide examples of such families. However, in all cases where
it could be verified, any limit of (classes of) such groups in ST happens to be a rank one
simple algebraic groups over a local field of positive characteristic. Indeed, T. Stulemeijer
has proved that if T is the regular tree of degree p + 1 with p prime, then the set of
isomorphism classes of algebraic groups in ST , denoted by STalg , is closed. Moreover the
set of non-isolated points are precisely the isomorphism classes of the simple algebraic
3

groups over local fields of positive characteristic. That set is finite (of cardinality 2) if
p > 2 and infinite if p = 2. We refer to [Stu16] for general results and full details.
Another potential source of examples for the implementation of that idea is the class
of complete Kac–Moody groups of rank two over finite fields. In that class, the tree T
is determined by the finite ground field. Letting the defining generalized Cartan matrix
run over the infinite set of possibilities in rank two, one obtains a countable family of
topologically simple boundary-2-transitive groups in Sub(Aut(T )). The difficulty arising
here is that we do not know whether those groups are pairwise non-isomorphic: we do not
even know whether they fall in infinitely many isomorphism classes. A discussion of this
rather subtle question, and partial answers, may be found in [Mar15, Theorem F and §6].
An important tool in the proofs of the results above is provided by the notion of kclosures recently introduced by Banks–Elder–Willis [BEW14], some of whose properties
are reviewed in §3 below. We establish a key relation between Chabauty convergence
and k-closures in the general context of automorphism groups of locally finite graphs, see
Proposition 3.2. We deduce the following statement, which is the main intermediate step
in the proof of Theorem 1.1.
Theorem 1.3. Let Λ be a locally finite connected graph and Γ ≤ Aut(Λ) act cocompactly
on Λ. Let Hn → H be a converging sequence in Sub(Aut(Λ)). Suppose that for each
n ≥ 1, there exists τn ∈ Aut(Λ) such that τn Γτn−1 ≤ Hn . Then we have
[H : H (∞) ] ≤ lim sup [Hn : Hn(∞) ].
n→∞

In particular, the set
{H ∈ Sub(Aut(Λ)) | H ≥ Γ and H = H (∞) }
is Chabauty-closed.
The condition that all groups Hn contain a conjugate of a fixed group Γ acting cocompactly may be viewed as a strengthening of the condition bounding the number of orbits,
which was imposed in Theorem 1.1. Classical results by Bass [Bas93] and Bass–Kulkarni
[BK90] ensure that when Λ is a tree, both conditions are equivalent (see §5.1 below).
Building upon this, we tighten the relation between Chabauty convergence of unimodular
cocompact subgroups of Aut(Λ) and k-closures (see Corollary 5.6) and deduce that the
algebraic properties of local pro-π-ness and local torsion-freeness are both Chabauty-open
in that context, see Propositions 5.19 and 5.22.
Taking advantage of the rather flexible hypotheses of Theorem 1.3, we include applications to groups acting on buildings that are not necessarily trees, see Corollary 6.3. We
are not aware of families of graphs other than trees where analogues of the aforementioned
results by Bass–Kulkarni hold. However, we note that chamber-transitive buildings whose
Weyl group is virtually free all admit a canonical continuous proper cocompact action on a
tree (see Lemma 6.6), so that the condition that the groups under consideration all contain
a conjugate of a fixed group Γ also becomes redundant in that context, see Corollary 6.8.
4

2

The Chabauty space

Given a locally compact group G, we denote by Sub(G) the set of closed subgroups of G
equipped with the Chabauty topology. Recall that a base of neighborhood of H ∈ Sub(G)
in the Chabauty topology is given by the sets
VK,U (H) := {J ∈ Sub(G) | J ∩ K ⊆ HU and H ∩ K ⊆ JU },
where K ranges over compact subsets of G and U over non-empty open subsets of G. When
G is metrizable, the convergence of a sequence in Sub(G) is characterized as follows.
Lemma 2.1. Let G be a locally compact metrizable group. A sequence (Hn ) in Sub(G)
converges to H ∈ Sub(G) if and only if the two conditions below are satisfied:
(i) Let (Hk(n) ) be a subsequence of (Hn ) and let (hk(n) ) be a sequence in G such that
hk(n) ∈ Hk(n) for each n ≥ 1. If (hk(n) ) converges to h ∈ G, then h ∈ H.
(ii) Any h ∈ H is the limit of a sequence (hn ) with hn ∈ Hn for each n ≥ 1.
Proof. See [GR06, Lemma 2].
The following results are then immediate.
Lemma 2.2. Let G be a locally compact metrizable group. The conjugation action of G
on Sub(G) is jointly continuous, i.e. if gn → g is a converging sequence in G and Hn → H
is a converging sequence in Sub(G), then gn Hn gn−1 → gHg−1 .
Proof. This is an easy consequence of Lemma 2.1.
Lemma 2.3. Let G be a locally compact metrizable group.
T
(1) If (Hn ) is a descending chain in Sub(G), then Hn → i≥1 Hi ;
(2) If (Hn ) is an ascending chain in Sub(G), then Hn →

S

i≥1 Hi .

Proof. WeTprove (1), the proof of (2) being similar. Let us check (i) and (ii) in Lemma 2.1.
Any h ∈ i≥1 Hi is the limit of the constant sequence (h), so (ii) is clear. Now in order
to prove (i), let hk(n) → h be a converging sequence in G such that hk(n) ∈ Hk(n) for each
n ≥ 1. For each i ≥ 1, the sequence (hk(n) )k(n)≥i is contained in Hi . Since
T Hi is closed
and hk(n) → h, we get h ∈ Hi . This being true for any i ≥ 1, we have h ∈ i≥1 Hi .
We also record the following essential result for the sake of future references.
Theorem 2.4. Let G be a locally compact group. The set Sub(G)0 of unimodular closed
subgroups of G is closed in Sub(G).
Proof. See [Bou63, Chapitre VIII, §5, no. 3, Théorème 1].
The next basic lemma plays a key role in the proof of Proposition 2.6.
5

Lemma 2.5. Let G be a locally compact group and C be a compact open subset of G (e.g.
C is a coset of a compact open subgroup). Then the set
{H ∈ Sub(G) | H ∩ C 6= ∅}
is clopen in Sub(G).
Proof. Since C is compact and open, there exists an identity neighborhood U in G such
that CU −1 = C. For any H ∈ Sub(G), we then consider the basic Chabauty-neighborhood
VC,U (H) = {J ∈ Sub(G) | J ∩ C ⊆ HU and H ∩ C ⊆ JU }
of H. We observe that, for any J ∈ VC,U (H), we have J ∩ C 6= ∅ if and only if H ∩ C 6= ∅.
Hence the set {H ∈ Sub(G) | H ∩ C 6= ∅} is clopen.
In the following proposition, as well as in the rest of the paper, we adopt the terminology from [Bas93, §1] concerning graphs. Given a graph Λ and a group H ≤ Aut(Λ)
acting without inversion on Λ, one can form the quotient graph H\Λ and the canonical
projection p : Λ → H\Λ. We recall from [BL01, §2.5] that the quotient graph Q = H\Λ is
an edge-indexed graph, i.e. it comes equipped with the map i associating to each edge
e ∈ E(Q) the cardinal
i(e) = #{a ∈ E(Λ) | p(a) = e and x is the origin of a},
where x is any vertex in Λ such that p(x) is the origin vertex of e. Since H always acts
without inversion on the first barycentric subdivision Λ(1) of Λ, it follows that any group
H ≤ Aut(Λ) yields a well defined edge-indexed quotient graph H\Λ(1) .
Proposition 2.6. Let Λ be a locally finite connected graph and (Q, i) be a finite edgeindexed graph. Then the following assertions hold.
(1) For any closed subset H ⊆ Sub(Aut(Λ)) consisting of groups acting without inversion,
the set
{H ∈ H | H\Λ ∼
= (Q, i)}
is clopen in H.
(2) The set

{H ∈ Sub(Aut(Λ)) | H\Λ(1) ∼
= (Q, i)}

is clopen in Sub(Aut(Λ)).
(3) For any C > 0, the set
Sub(Aut(Λ))≤C := {H ∈ Sub(Aut(Λ)) | #V (H\Λ) ≤ C}
is clopen in Sub(Aut(Λ)).

6

Proof. (1). Let F ⊆ V (Λ) ∪ E(Λ) be a finite set of vertices and edges of Λ. We denote by
Hco-F
the set of those H ∈ H such that HF = V (Λ) ∪ E(Λ), i.e. those H ∈ H such that F meets
every H-orbit of vertices and every H-orbit of edges in Λ.
Claim 1. The set Hco-F is closed in H.
Proof of the claim: Let Hn → H be a converging sequence in H such that Hn ∈ Hco-F
for all n. Pick any x ∈ V (Λ) ∪ E(Λ). Then for all n there exists hn ∈ Hn with hn (x) ∈
F. Since F is finite, the sequence (hn ) is bounded, hence subconverges to an element
h ∈ Aut(Λ) satisfying h(x) ∈ F. By Lemma 2.1 (i) we have h ∈ H. This confirms that
HF = V (Λ) ∪ E(Λ).

Claim 2. The set Hco-F is open in H.
Proof of the claim: Define Fe as the set consisting of all vertices that are adjacent to a
vertex in F or incident to an edge in F. Since F is finite and Λ is locally finite, we infer
that Fe is finite. Given any H ∈ Hco-F , for any x ∈ Fe there exists hx ∈ H with hx (x) ∈ F.
Since Fe is finite, Lemma 2.5 ensures that the set
{J ∈ Sub(Aut(Λ)) | ∀x ∈ Fe, J ∩ hx Aut(Λ)x 6= ∅}

is an open set containing H, where Aut(Λ)x denotes theSstabilizer of x in Aut(Λ). If J is
any group belonging to that set, we observe that X = g∈J gF is a subset of the graph
Λ satisfying the property that for any vertex x in X, all edges of Λ incident to x and all
vertices of Λ adjacent to x are also in X. Since Λ is connected, we deduce that X = Λ.
Therefore, if in addition J ∈ H, then J ∈ Hco-F . The claim follows.

Claim 3. If (Q, i) is a finite edge-indexed graph, then the set
V(Q,i),F := {H ∈ Hco-F | H\Λ ∼
= (Q, i)}
is clopen in H.
Proof of the claim: For each H ∈ Hco-F , the isomorphism type of the edge-indexed quotient
graph H\Λ is completely determined by the finite set
SH := {(x, y) ∈ F × Λ | ∃h ∈ H : hx = y and d(y, F) ≤ 1}.
Moreover, it is clear from Lemma 2.1 that if Hn → H in Hco-F then SHn = SH for
sufficiently large n. Consequently, the set V(Q,i),F is clopen in Hco-F . As Hco-F is itself
clopen in H by the first two claims, the conclusion follows.

We now finish the proof as follows. Given a finite edge-indexed graph (Q, i), we must
show that the set V(Q,i) := {H ∈ H | H\Λ ∼
= (Q, i)} is clopen in H. We may assume that
it is nonempty. Fix a base vertex v0 ∈ V (Λ). For any group H ∈ V(Q,i) , we can find a set of
representatives F0 of the H-orbits of vertices and edges in Λ, in such a way that v0 ∈ F0
7

and that F0 is connected. Notice that there are only finitely many connected subsets
F ⊆ V (Λ) ∪ E(Λ) containing v0 and such that #V (F) = #V (Q) and #E(F)
S = #E(Q).
Let us enumerate all of them, namely F0 , F1 , . . . , Fm . We have V(Q,i) = m
j=0 V(Q,i),Fj .
Each V(Q,i),Fj is clopen by the third claim, hence V(Q,i) is clopen as well.
(2). We may identify Aut(Λ) with a closed subgroup of Aut(Λ(1) ). Under that identification, the set Sub(Aut(Λ)) is identified with a closed subset H of Sub(Aut(Λ(1) ))
consisting of groups acting without inversion. The desired assertion then follows from (1).
(3). For any C > 0, there exists C ′ > 0 such that any H ∈ Sub(Aut(T )) satisfying
#V (H\Λ) ≤ C also satisfies #V (H\Λ(1) ) ≤ C ′ . The conclusion then follows from (2),
since there are finitely many edge-indexed graphs (Q, i) with #V (Q) ≤ C ′ and such that
i(e) is bounded above by the maximal degree of a vertex in Λ(1) for all e ∈ E(Q).

3

The k-closure of a graph automorphism group

Let Λ be a locally finite connected graph with vertex set V (Λ). We define the k-closure
J (k) of an automorphism group J ≤ Aut(Λ) by
J (k) = {g ∈ Aut(Λ) | ∀v ∈ V (Λ), ∃h ∈ J : g|B(v,k) = h|B(v,k) },
where B(v, k) is the ball centered at v and of radius k in Λ. That notion was first
introduced and studied by Banks–Elder–Willis in [BEW14], in the case where Λ is a tree.
It is clear from the definition that J (k) ⊇ J (ℓ) ⊇ J for any k ≤ ℓ. Other basic properties
of k-closures, due to Banks–Elder–Willis, are collected in the following.
Lemma 3.1. Let Λ be a locally finite connected graph. For any k ≥ 0 and J ≤ Aut(Λ),
J (k) is a closed subgroup of Aut(Λ). Moreover we have
\
J=
J (k) .
k≥0

Proof. The proofs when Λ is a locally finite tree are given in [BEW14, Proposition 3.4],
but they are independent from the tree structure and thus also work for any locally finite
connected graph Λ.
In view of Lemma 2.3 (1), the previous lemma implies that J (k) → J in Sub(Aut(Λ)).
The next result is then a key tool for the proof of Theorem 1.3. In order to facilitate its
statement, we introduce the following notation. Given a group Γ ≤ Aut(Λ), we write
Sub(Aut(Λ))≥Γ = {H ∈ Sub(Aut(Λ)) | H ≥ τ Γτ −1 for some τ ∈ Aut(Λ)}.
Observe that if the normalizer of Γ in Aut(Λ) is cocompact, then Sub(Aut(Λ))≥Γ is
Chabauty-closed. Given a group H ≤ Aut(Λ), a vertex v ∈ V (Λ) and an integer r ≥ 0,
[r]
we also write Hv for the pointwise stabilizer of the ball B(v, r) in H.

8

Proposition 3.2. Let Λ be a locally finite connected graph, Γ ≤ Aut(Λ) act cocompactly
on Λ and H ∈ Sub(Aut(Λ))≥Γ . Fix v0 ∈ V (Λ). Then for all k ≥ 0, the set
Vk := {J ∈ Sub(Aut(Λ))≥Γ | σJσ −1 ≤ H (k) for some σ ∈ Aut(Λ)v[k]0 }
is a neighborhood of H in Sub(Aut(Λ))≥Γ .
Proof. Consider a sequence Hn → H in Sub(Aut(Λ))≥Γ and let us show that Hn ∈ Vk
for sufficiently large n. Let X ⊂ Λ be a compact fundamental domain for the action of
Γ on Λ. For each n, let τn ∈ Aut(Λ) be such that Hn ≥ τn Γτn−1 . We may assume, up to
precomposing τn with an adequate element of Γ, that τn sends v0 to a vertex in X. Since
X is compact, the sequence (τn ) is bounded and we can further assume (by passing to a
subsequence) that (τn ) converges to some τ ∈ Aut(Λ). Define σn := τ τn−1 for each n ≥ 1
so that σn → id. In this way, we have
Γ′ := τ Γτ −1 ≤ τ τn−1 Hn τn τ −1 = σn Hn σn−1 =: Hn′

for each n ≥ 1,

where Γ′ also acts cocompactly on Λ with X ′ := τ (X) as a fundamental domain. Moreover,
as σn → id, we have Hn′ → H by Lemma 2.2 and in particular Γ′ ≤ H.
In order to conclude, it suffices to find N ≥ 1 such that Hn′ ≤ H (k) for each n ≥ N .
Let D be the diameter of X ′ and set
K := {g ∈ Aut(Λ) | d(g(v0 ), v0 ) ≤ 2D}
and


U := g ∈ Aut(Λ) | g|B(v0 ,k+D) = id |B(v0 ,k+D) .

The set K is compact and the set U is open, so there exists N ≥ 1 such that Hn′ ∈ VK,U (H)
for each n ≥ N . In particular, we have Hn′ ∩ K ⊆ HU for each n ≥ N . This exactly
means that, for any g ∈ Hn′ (with n ≥ N ) satisfying d(g(v0 ), v0 ) ≤ 2D, there exists h ∈ H
such that g|B(v0 ,k+D) = h|B(v0 ,k+D).
We need to show that Hn′ ≤ H (k) for each n ≥ N . In order to do so, consider g ∈ Hn′
with n ≥ N and v ∈ V (Λ). Let γ1 ∈ Γ′ be such that d(γ1 g(v), v0 ) ≤ D and γ2 ∈ Γ′
be such that d(γ2 (v0 ), v) ≤ D. Those elements exist because D is the diameter of the
fundamental domain X ′ for the action of Γ′ . The two previous inequalities imply that
d(γ1 gγ2 (v0 ), v0 ) ≤ 2D. Hence, by definition of N there exists h ∈ H with
γ1 gγ2 |B(v0 ,k+D) = h|B(v0 ,k+D) ,
which is equivalent to saying that
g|B(γ2 (v0 ),k+D) = γ1−1 hγ2−1 |B(γ2 (v0 ),k+D) .
But d(γ2 (v0 ), v) ≤ D, so B(γ2 (v0 ), k + D) ⊇ B(v, k) and
g|B(v,k) = γ1−1 hγ2−1 |B(v,k) ,
which is sufficient to conclude since γ1−1 hγ2−1 ∈ H.
9

The following observation describes a local algebraic property that is preserved when
taking the k-closure (with a sufficiently large k).
Proposition 3.3. Let Λ be a locally finite connected graph and H ∈ Sub(Aut(Λ)). Let
[r]
also π be a set of primes and r ≥ 0. Suppose that Hv is a pro-π group for all v ∈ V (Λ).
[r]
Then for each k ≥ r + 1, the group (H (k) )v is a pro-π group for all v ∈ V (Λ).
In particular, if H acts cocompactly on Λ and has an open pro-π subgroup, then so
does H (k) for all sufficiently large k.
Proof. Since H (k) ≤ H (ℓ) for all k ≥ ℓ and since a closed subgroup of a pro-π group is
pro-π, it suffices to consider G = H (r+1) . We show that for each n ≥ r and each v ∈ V (Λ),
[n] 
is a π-group. This assertion implies the required conclusion.
the finite group Gv G[n+1]
v
[n] 
is not a
Fix n ≥ r and v ∈ V (Λ) and assume for a contradiction that Gv G[n+1]
v
π-group. Then it contains an element g of prime order p, with p 6∈ π. There exists a
vertex z with d(v, z) = n such that the restriction g|B(z,1) contains a p-cycle. Let x be a
vertex on a geodesic path from v to z such that d(x, z) = r. Thus g fixes B(x, r) ⊆ B(v, n)
pointwise. Since g ∈ G = H (r+1) , there is h ∈ H such that g|B(x,r+1) = h|B(x,r+1) . Hence
[r]

[r+1]

h belongs to Hx and the image of h modulo Hx
is of order p 6∈ π. This contradicts
[r]
the hypothesis that Hx is pro-π.
Now suppose H acts cocompactly on Λ and has an open pro-π subgroup U . Since U
[r]
is open, there exists v0 ∈ V (Λ) and r ≥ 0 such that Hv0 ⊆ U . Let X ∋ v0 be a compact
fundamental domain for the action of H on Λ, and denote by D its diameter. For each
vertex x ∈ X, we have
Hx[r+D] ⊆ Hv[r]
⊆ U,
0
[r+D]

so Hx
is a pro-π group. Since X is a fundamental domain for the action of H on Λ,
[r+D]
we even have that Hv
is a pro-π group for all v ∈ V (Λ). By the previous assertion,
(k)
this implies that H has an open pro-π subgroup for each k ≥ r + D + 1.
Applying the previous proposition in the case of the empty set of primes, we obtain
the following corollary for discrete groups.
Corollary 3.4. Let Λ be a locally finite connected graph and H be a discrete subgroup of
Aut(Λ) acting cocompactly on Λ. Then H = H (k) for all sufficiently large k.
Proof. Applying Proposition 3.3 to the empty set π = ∅, we obtain that H (k) is discrete for
each sufficiently large k. Since H acts cocompactly on Λ, so does H (k) for any k. Fixing
k0 such that H (k0 ) is discrete, we deduce that the index of H in H (k0 ) is finite. Since
H ≤ H (k+1) ≤ H (k) ≤ H (k0 ) for each k ≥ k0 , the conclusion follows from Lemma 3.1.

4

Finite quotients of groups acting on graphs

The goal of this section is to prove Theorem 1.3.

10

We first recall that, for a topological group G, the symbol G(∞) denotes the intersection
of all open subgroups of finite index of G. The following lemma is classical. The notation
P ≤of i G means that P is an open subgroup of finite index of G.
Lemma 4.1. Let G be a topological group and P ≤of i G. There exists R ≤ P such that
R Eof i G. In particular, G(∞) coincides with the intersection of all open normal subgroups
of finite index of G.
Proof. It suffices to take for R the kernel of the natural action of G on G/P .
The next result shows, in the context of automorphism groups of graphs, how k-closures
preserve open subgroups of finite index.
Lemma 4.2. Let Λ be a locally finite connected graph and H ∈ Sub(Aut(Λ)) act cocompactly on Λ. If P ≤of i H, then [H (k) : P (k) ] ≤ [H : P ] for all sufficiently large k.
Proof. Fix v0 ∈ V (Λ) and let m = [H : P ]. We can write
H=

m
G

hi P

i=1

for some h1 , . . . , hm ∈ H. Since H acts cocompactly on Λ, the action of P on Λ is also
cocompact. Let X ⊆ Λ be a compact fundamental domain for the action of P and denote
by D the diameter of X.
The fact that P is an open subgroup of H implies that there exists R ≥ 0 with
Hv[R]
⊆ P.
0
We claim that [H (k) : P (k) ] ≤ m for each k ≥ R + D + 1. To prove the claim, we fix
k ≥ R + D + 1 and show that
m
[
(k)
hi P (k) .
H =
i=1

Take g ∈ H (k) and v ∈ V (Λ). There exists i ∈ {1, . . . , m} and x ∈ P such that g|B(v,k) =
hi x|B(v,k) , which is equivalent to saying that h−1
i g|B(v,k) = x|B(v,k) . If we prove that i is
(k) which will end the proof.
independent of the choice of v, then we will get h−1
i g ∈ P
Since Λ is connected, it suffices to show that the value of i is the same for any two adjacent
vertices. Fix v and v ′ two neighboring vertices of Λ and suppose that
g|B(v,k) = hi x|B(v,k)

and g|B(v′ ,k) = hj y|B(v′ ,k)

for some x, y ∈ P and some i, j ∈ {1, . . . , m}. It follows that
hi x|B(v,k−1) = hj y|B(v,k−1)
or equivalently that
−1
h−1
j hi xy |B(y(v),k−1) = id |B(y(v),k−1) .

11

[k−1]

−1 is thus such that e ∈ H
The element e := h−1
j hi xy
y(v) . As X is a fundamental domain
(with diameter D) for the action of P on Λ, there exists p ∈ P such that p(y(v)) ∈ B(v0 , D).
Hence, the element pep−1 satisfies
[k−1]

⊆ Hv[R]
⊆ P.
pep−1 ∈ Hp(y(v)) ⊆ Hv[k−1−D]
0
0
−1 = e ∈ P and thus h−1 h ∈ P , which implies that i = j as desired.
We get h−1
i
j hi xy
j

Before proving Theorem 1.3, we still need a technical lemma.
Lemma 4.3. Let Λ be a locally finite connected graph and Hn → H, Ln → L be two
converging sequences in Sub(Aut(Λ)) such that Ln ≤ Hn for each n ≥ 1. Assume that
there exists C > 0 such that Hn ∈ Sub(Aut(Λ))≤C for each n ≥ 1. Suppose also that
there exists S ≥ 1 such that [Hn : Ln ] ≤ S for each n ≥ 1. Then L ≤ H and [H : L] ≤ S.
Proof. The fact that L ≤ H is clear. For each n ≥ 1, let Fn ⊆ Aut(Λ) be such that
Hn = Ln Fn and |Fn | ≤ S. We directly get that Ln ∈ Sub(Aut(Λ))≤CS for each n ≥ 1.
If v0 ∈ V (Λ) is a fixed vertex, for each n ≥ 1 and f ∈ Fn we can thus assume that
d(f (v0 ), v0 ) ≤ CS. By adding elements to Fn if necessary, we can also suppose that
(n)
(n)
|Fn | = S and write Fn = {f1 , . . . , fS }. Since the set {g ∈ Aut(Λ) | d(g(v0 ), v0 ) ≤ CS}
(n)
is compact, we can finally assume by passing to subsequences that (fi ) converges to some
fi ∈ Aut(Λ) for each i ∈ {1, . . . , S}. Define F := {f1 , . . . , fS }. It is clear that F ⊆ H and
we claim that H = LF . Take h ∈ H. By Lemma 2.1, there exists a converging sequence
(n)
hn → h with hn ∈ Hn for each n ≥ 1. As Hn = Ln Fn , we can write hn = ℓn fin with
ℓn ∈ Ln and in ∈ {1, . . . , S}. There is a subsequence (ik(n) ) of (in ) which is constant, say
(n)

(n)

equal to j ∈ {1, . . . , S}. Then hk(n) = ℓk(n) fj and hence ℓk(n) = hk(n) (fj )−1 → hfj−1 .
It is also true that this limit belongs to L, so hfj−1 = ℓ ∈ L and h = ℓfj .
The proof of Theorem 1.3 is now an easy combination of the previous results.
(∞)

Proof of Theorem 1.3. Let S = lim supn→∞ [Hn : Hn ]. Without loss of generality, we
(∞)
may assume that [Hn : Hn ] ≤ S for each n ≥ 1. By Proposition 3.2, we may further
assume that for each k ≥ 0, there exists N (k) ≥ 1 such that Hn ≤ H (k) for each n ≥ N (k).
In order to prove that [H : H (∞) ] ≤ S, it suffices to prove that [H : P ] ≤ S for each
P ≤of i H. By Lemma 4.2, there exists K ≥ 0 such that P (k) ≤of i H (k) for any k ≥ K.
Let us temporarily fix k ≥ K. For each n ≥ N (k), we have Hn ≤ H (k) and hence
P (k) ∩ Hn ≤of i Hn . By hypothesis, this means that [Hn : P (k) ∩ Hn ] ≤ S. Letting n tend
to infinity, we obtain with Lemma 4.3 that [H : P (k) ∩ H] ≤ S for each k ≥ K. Now letting
k tend to infinity and because P (k) → P (see Lemmas 2.3 (1) and 3.1), we get [H : P ] ≤ S.
An open subgroup is always closed, so P = P and the conclusion follows.

12

5
5.1

Trees
Existence and conjugation of tree lattices

When Λ is a locally finite tree, Propositions 5.1 and 5.3 below (which come from [BK90]
and [Bas93] respectively) can be used to drop the hypothesis about Γ in Theorem 1.3.
Proposition 5.1. Let T be a locally finite tree. Let H ≤ Aut(T ) act cocompactly on T
and suppose that H is unimodular. Then H contains a free uniform lattice, i.e. there
exists a discrete subgroup Γ ≤ H acting freely and cocompactly on T .
Proof. See [BK90, Existence Theorem].
Given a tree T and two groups H, H ′ ≤ Aut(T ) acting without inversion on T , we
write the equality
H\T = H ′ \T
whenever H and H ′ have the same orbits on T . The latter condition, which means that
the canonical projections p : T → H\T and p′ : T → H ′ \T coincide, implies in particular
that the quotient graphs H\T and H ′ \T are isomorphic as edge-indexed graphs, since the
edge-indexing function of the quotient graph is completely determined by the projection
map. The following basic fact clarifies the difference between isomorphism and equality
of quotients.
Lemma 5.2. Let T be a tree and H, H ′ ≤ Aut(T ) act without inversion on T . If H\T
and H ′ \T are isomorphic as edge-indexed graphs, then there exists g ∈ Aut(T ) such that
H\T = H ′′ \T , where H ′′ = gH ′ g −1 .
Proof. This follows from [Bas93, Remark 1.18].
Proposition 5.3. Let T be a tree and H, H ′ ≤ Aut(T ) act without inversion on T .
Suppose that H\T = H ′ \T . If Γ ≤ H acts freely on T , then there exists τ ∈ Aut(T ) such
that τ Γτ −1 ≤ H ′ .
Proof. See [Bas93, Corollary 5.3].
Corollary 5.4. Let T be a locally finite tree and H ∈ Sub(Aut(T )). Suppose that H is
unimodular and acts cocompactly on T (these conditions hold, for instance, if H is edgetransitive and type-preserving). Then H has a subgroup Γ acting freely and cocompactly
on T such that Sub(Aut(T ))≥Γ is a neighborhood of H in Sub(Aut(T )).
Proof. Upon replacing T by its first barycentric subdivision, we may assume that H acts
without inversion. By Proposition 5.1, there exists Γ ≤ H acting cocompactly and freely
on T . Consider a converging sequence Hn → H in Sub(Aut(T )). By Proposition 2.6 (3)
Hn acts with at most C orbits of vertices for all sufficiently large n, where C = #V (H\T ).
We may then deduce from Lemma 2.5 that Hn acts without inversion for all sufficiently
large n because H does. Moreover Proposition 2.6 (1) ensures that, for sufficiently large
n, the quotient graphs Hn \T and H\T are isomorphic as edge-indexed graphs. Hence,
by Lemma 5.2 and Proposition 5.3, for sufficiently large n, there exists τn ∈ Aut(T ) such
that τn Γτn−1 ≤ Hn , i.e. Hn ∈ Sub(Aut(T ))≥Γ .
13

The following result shows that in Corollary 5.4, the choice of Γ can be made uniform,
i.e. independent of the choice of H. In order to make this precise, we define
Sub(Aut(T ))0≤C := {H ∈ Sub(Aut(T ))≤C | H is unimodular}.
Corollary 5.5. Let T be a locally finite tree. For each C > 0, the set Sub(Aut(T ))0≤C is
clopen in Sub(Aut(T )). Moreover there exists a subgroup Γ ≤ Aut(T ) acting freely and
cocompactly on T such that
Sub(Aut(T ))0≤C ⊆ Sub(Aut(T ))≥Γ .
Proof. We already know by Proposition 2.6 (3) that Sub(Aut(T ))≤C is a clopen subset
of Sub(Aut(T )). Moreover the set Sub(Aut(T ))0 of unimodular subgroups is closed by
Theorem 2.4. In particular Sub(Aut(T ))0≤C is closed.
For each H ∈ Sub(Aut(T ))0≤C , Corollary 5.4 yields a discrete cocompact group Γ
such that Sub(Aut(T ))≥Γ is a neighborhood of H. Since every locally compact group
containing a lattice is unimodular, this implies that Sub(Aut(T ))0≤C is also open.
Let us now partition the set Sub(Aut(T ))0≤C into subsets V1 , . . . , Vm in such a way
that H, H ′ ∈ Vi if and only if H\T ∼
= H ′ \T as edge-indexed graphs. By Lemma 5.2
and Proposition 5.3, for each i there exists a discrete cocompact group Γi such that
Vi ⊆ Sub(Aut(T ))≥Γi . In particular
Sub(Aut(T ))0≤C

⊆

m
[

Sub(Aut(T ))≥Γi .

i=1

By [BK90, Commensurability Theorem], upon replacing each Γi by a conjugate, we may
assume that they are pairwise commensurate,
i.e. the index of Γi ∩ Γj is of finite index in
Tm
Γi for all i and j. It follows that Γ = i=1 Γi is itself a cocompact lattice in Aut(T ). The
required assertion follows since
Sub(Aut(T ))≥Γi ⊆ Sub(Aut(T ))≥Γ
for all i.
We then deduce the following corollary from Proposition 3.2.
Corollary 5.6. Let T be a locally finite tree and H ∈ Sub(Aut(T )). Suppose that H is
unimodular and acts cocompactly on T . Fix v0 ∈ V (T ). Then for each k ≥ 0, the set
{J ∈ Sub(Aut(T )) | σJσ −1 ≤ H (k) for some σ ∈ Aut(T )v[k]0 }
is a neighborhood of H in Sub(Aut(T )).
Proof. This is the combination of Corollary 5.4 and Proposition 3.2.
Arguing similarly, we obtain the following consequence of Theorem 1.3.
14

Corollary 5.7. Let T be a locally finite tree and Hn → H be a converging sequence in
Sub(Aut(T )). Suppose that H is unimodular and acts cocompactly on T . Then we have
[H : H (∞) ] ≤ lim sup [Hn : Hn(∞) ].
n→∞

In particular, if Hn has no proper open subgroup of finite index for each n ≥ 1 then H
has no proper open subgroup of finite index.
Proof. Corollary 5.4 ensures that Hn ∈ Sub(Aut(T ))≥Λ for sufficiently large n. The
conclusion then follows from Theorem 1.3.

5.2

Limits of simple groups acting on trees

The goal of this section is to prove the next theorem, which is a stronger version of
Theorem 1.1.
Theorem 5.8. Let T be a locally finite tree all of whose vertices have degree ≥ 2. For
any C > 0, the Chabauty-closure of the set of abstractly simple groups in Sub(Aut(T ))≤C
is the set of groups in Sub(Aut(T ))≤C without proper open subgroup of finite index.
We start by proving the following.
Proposition 5.9. Let T be a locally finite tree and let C > 0. The set
(∞)

Sub(Aut(T ))≤C := {H ∈ Sub(Aut(T ))≤C | H = H (∞) }
is closed in Sub(Aut(T )).
(∞)

Proof. Let Hn → H be a converging sequence in Sub(Aut(T )) with Hn ∈ Sub(Aut(T ))≤C
for each n. We already know by Proposition 2.6 (3) that H ∈ Sub(Aut(T ))≤C . For each
n ≥ 1, Hn acts cocompactly on T and is thus compactly generated. Therefore, the image of
the modular character of Hn is a finitely generated subgroup of R, which is thus residually
(∞)
finite. In particular, the condition that Hn = Hn implies that Hn is unimodular. By
Theorem 2.4, H is also unimodular and we can apply Corollary 5.7 to get H = H (∞) , as
(∞)
required. This confirms that Sub(Aut(T ))≤C is closed.
(∞)

There remains to show that any group in Sub(Aut(T ))≤C is a limit of abstractly
simple groups in that same set. Before proving this we need two more technical results.
Lemma 5.10. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and H ≤
Aut(T ) be a closed subgroup without any infinite cyclic discrete quotient (e.g. H = H (∞) ).
If H acts cocompactly on T , then it does not preserve any proper non-empty subtree and
does not fix any end of T .

15

Proof. Since all vertices of T have degree ≥ 2 and H acts cocompactly on T , we deduce
from [Tit70, Lemme 4.1] that H does not preserve any non-empty subtree of T .
Suppose now for a contradiction that H fixes some end b ∈ ∂T . Let (vn ) be a ray
in T toward b. Then the map φ : H → Z defined by φ(h) := limn→∞ d(h(vn ), vn ) is a
group homomorphism and has infinite image (because H acts cocompactly on T ), which
contradicts the fact that H has no infinite cyclic discrete quotient.
In the following proposition and as in [BEW14], given J ≤ Aut(T ) and k > 0, the
symbol J +k denotes the subgroup of J generated by the pointwise stabilizers of (k−1)-balls
around edges of T .
Proposition 5.11. Let T be a locally finite tree and G ≤ Aut(T ) be a non-discrete group
which acts cocompactly on T , does not preserve any proper non-empty subtree and does
not fix any end of T . Suppose that G = G(k) for some k ≥ 0. Then G+k is abstractly
simple and G/G+k is virtually free.
Proof. From [BEW14, Theorem 7.3] we know that G+k is abstractly simple or trivial.
Also, it is clear from the definition that G+k is an open normal subgroup of G. Since G is
non-discrete, G+k is non-discrete and in particular non-trivial (hence simple).
The discrete quotient group G/G+k acts cocompactly on the quotient graph G+k \T .
Bass–Serre theory ensures that G+k is the fundamental group of a graph of groups, whose
underlying graph is nothing but G+k \T (see [Ser77, §I.5.4, Théorème 13]). Since G+k is
simple and non-discrete, it is generated by the vertex stabilizers. It then follows that the
quotient graph G+k \T is a tree (see [Ser77, §I.5.4, Corollaire 1]). We next observe that
the G/G+k -action on the tree G+k \T is proper. Indeed, a coset gG+k stabilizes a vertex in
G+k \T if and only if gv ∈ G+k v for some v ∈ V (T ). This is equivalent to the requirement
that g ∈ G+k U , where U is the stabilizer of v in G, which is compact. This confirms
that the stabilizer of a vertex of G+k \T in the discrete quotient group G/G+k is indeed
compact, hence finite. Therefore G/G+k is a discrete group acting properly cocompactly
on a tree. It is thus virtually free.
Proposition 5.12. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and
(∞)
let C > 0. In Sub(Aut(T ))≤C , the subset consisting of the abstractly simple groups is
dense.
(∞)

Proof. Pick any H ∈ Sub(Aut(T ))≤C . We must show that H is a limit of abstractly
(∞)

simple groups contained in Sub(Aut(T ))≤C . For each k > 0, set Hk = (H (k) )+k . First
note that H is not discrete, otherwise it would be virtually free, hence residually finite,
contradicting H = H (∞) . We can therefore invoke Lemma 5.10 and Proposition 5.11
(applied to H (k) ) to get that Hk is abstractly simple and H (k) /Hk is virtually free.
Since H (k) /Hk is virtually free, it is residually finite. Recalling now that H has no
finite discrete quotient other than the trivial one, we infer that H hasTtrivial image in
H (k) /Hk , so that H ≤ Hk ≤ H (k) . Lemma 3.1 then implies that H = k≥1 Hk , so that
Hk → H by Lemma 2.3 (1), thereby completing the proof.

16

Proof of Theorem 5.8. Follows by assembling Propositions 5.9 and 5.12.
Remark 5.13. It is important to note that the set
{H ∈ Sub(Aut(T )) | H is locally 2-transitive and H = H (∞) }
(in the terminology of [BM00]) may contain groups that are not topologically simple.
Explicit examples of such H are constructed in [BM00, Example 1.2.1], where T is regular
of degree p2 + p + 1 (p being an arbitrary prime). In particular, the set of topologically
simple locally 2-transitive closed subgroups of Aut(T ) is generally not Chabauty-closed.
The following result shows that the conclusion of Theorem 1.1 may fail if the tree T
is allowed to have vertices of degree 1.
Lemma 5.14. Let T be the universal covering tree of the graph on 7 vertices depicted in
the figure below. Let V1 , V3 and V8 denote the set of vertices of T of degree 1, 3 and 8
respectively. Let X be the subtree of T which is the convex hull of V3 . Thus X is isomorphic
to the trivalent tree. Its vertex set is V3 ∪ V8 , and those two sets V3 and V8 are the two
parts in the canonical bipartition of X. The following assertions hold.
(1) Aut(T ) has a closed subgroup H isomorphic to


Y

Alt(5) ⋊ Aut(X)+ .
v∈V8

(2) H = H (∞) .
(3) H is not a Chabauty limit of topologically simple closed subgroups of Aut(T ).
b

b

b

b

b

b

b

Proof. (1) The subtree X is Aut(T )-invariant. Thus we have a canonical continuous
homoQ
morphism Aut(T ) → Aut(X). Its kernel is compact and
isomorphic
to
Sym(5).
v∈V8
Q
It contains a characteristic subgroup K isomorphic to v∈V8 Alt(5). Moreover Aut(T )
has a closed subgroup S isomorphic to Aut(X)+ . The requested subgroup H can be
defined as H = KS.
17

(2) Let N be an open normal subgroup of finite index in H = KS. Then N ∩Q
K is an open
normal subgroup
of
K,
and
thus
contains
all
but
finitely
many
factors
of
v∈V8 Alt(5)
Q
(i.e. N ∩ K ⊇ v∈I Alt(5) for some cofinite set I ⊆ V8 ). Since the conjugation action
of H is transitive on those factors, we infer that N contains them all. Hence K ≤ N .
Thus the quotient map H → H/N factors through H/K ∼
= S, which is topologically
simple. Hence H/N is trivial, which confirms that H = H (∞) .
(3) Any infinite topologically simple subgroup of Aut(T ) acts faithfully on X. On the
other hand, the group H contains an element h fixing a vertex v ∈ V8 and permuting
cyclically the 5 neighbors of v with degree 1. Any closed subgroup J of Aut(T ) which
is sufficiently close to H in the Chabauty topology also contains elements fixing v
with the same action on its neighbors. In particular the stabilizer Jv has a non-trivial
5-Sylow subgroup. Since every vertex stabilizer in Aut(X)+ is a pro-{2, 3} group, we
deduce that J does not act faithfully on X and is thus not topologically simple.

5.3

Boundary-2-transitive automorphism groups of trees

Recall that the monolith Mon(G) of a topological group G is defined to be the (possibly trivial) intersection of all its non-trivial closed normal subgroups. It is clear from
Lemma 4.1 that, when G is infinite, Mon(G) ≤ G(∞) (because an open subgroup is always
closed). If moreover G is totally disconnected and locally compact, then it appears that
Mon(G) = G(∞) as soon as Mon(G) is cocompact in G.
Lemma 5.15. Let G be a totally disconnected locally compact group. If G/ Mon(G) is
compact, then G(∞) ≤ Mon(G).
Proof. By Lemma 4.1, we have

\

G(∞) =

N.

N Eof i G

The group G/ Mon(G) is compact by hypothesis and totally disconnected (as a quotient
of a totally disconnected locally compact group by a closed subgroup), so it is profinite. In
particular, the open (and hence finite index) normal subgroups of G/ Mon(G) form a base
of neighborhood of the identity. Their intersection is thus trivial, which implies that the
intersection of all open normal subgroups of finite index of G is contained in Mon(G).
The previous lemma can be applied when G is a boundary-2-transitive automorphism
group of a tree, as the following result (due to M. Burger and S. Mozes) shows.
Proposition 5.16. Let T be a locally finite thick tree and H ∈ Sub(Aut(T )) act 2transitively on ∂T . Then H/ Mon(H) is compact and Mon(H) is topologically simple. In
particular Mon(H) = H (∞) .
Proof. This follows from [BM00, Propositions 1.2.1 and 3.1.2, Lemma 3.1.1] (see also
Proposition 6.2 below).

18

Corollary 5.17. Let T be a locally finite thick tree and Hn → H be a converging sequence
in Sub(Aut(T )) whose limit H acts 2-transitively on ∂T . Then we have
[H : Mon(H)] ≤ lim sup [Hn : Hn(∞) ].
n→∞

In particular, if Hn has no proper open subgroup of finite index for each n ≥ 1 then H
is topologically simple.
Proof. This follows by assembling Corollary 5.7 and Proposition 5.16.
Corollary 5.18. Let T be a locally finite thick tree. The set of topologically simple closed
subgroups of Aut(T ) acting 2-transitively on ∂T is closed in Sub(Aut(T )).
Proof. It follows easily from [BM00, Lemma 3.1.1] that the set of boundary-2-transitive
groups is closed in Sub(Aut(T )) (and is contained in Sub(Aut(T ))≤2 ). Within that set,
the subset of topologically simple groups is closed in view of Corollary 5.17.
Recall that, for a locally finite thick tree T , we defined the space ST by
ST := {H ∈ Sub(Aut(T )) | H is topologically simple and 2-transitive on ∂T } /∼
=,
where ∼
= is the relation of topological isomorphism. In our context, it actually appears
that two groups are topologically isomorphic if and only if they are conjugate in Aut(T )
(see [Rad15, Proposition B.1]). This equivalence enables us to show Theorem 1.2.
Proof of Theorem 1.2. The first assertion is exactly Corollary 5.18. Also, by [Rad15,
Proposition B.1], the isomorphism classes in the set
{H ∈ Sub(Aut(T )) | H is topologically simple and 2-transitive on ∂T }
coincide with the conjugacy classes. We now prove that they are closed. Let σn Hσn−1 → H ′
be a sequence of groups in that set which are conjugate to H. As H is edge-transitive,
we can assume that σn sends a fixed vertex v0 to a vertex at distance ≤ 1 for each n ≥ 1.
Hence, (σn ) subconverges to some σ ∈ Aut(T ) and H ′ = σHσ −1 by Lemma 2.2.

5.4

Local prime content and local torsion-freeness

Let T be a locally finite tree. In this section, we provide applications of Corollary 5.6 by
highlighting two algebraic properties that define open subsets of the space
Sub(Aut(T ))0≤C := {H ∈ Sub(Aut(T ))≤C | H is unimodular}.
Let π be a set of primes. A totally disconnected locally compact group is called locally
pro-π if it has an open pro-π subgroup. If G is the full automorphism group of a regular
rooted tree, then the set of locally pro-π subgroups is generally neither open nor closed
in the Chabauty space Sub(G). The following result shows that this situation changes if
one considers closed subgroups of bounded covolume in Aut(T ).
19

Proposition 5.19. Let T be a locally finite tree all of whose vertices have degree ≥ 2
and let C > 0. Then for any set of primes π, the set of locally pro-π groups is open in
Sub(Aut(T ))0≤C . In particular the set of discrete subgroups is open in Sub(Aut(T ))0≤C .
Proof. Let H be a locally pro-π group in Sub(Aut(T ))0≤C . By Proposition 3.3, there exists
K ≥ 0 such that H (K) is also locally pro-π. We also know from Corollary 5.6 that the set
{J ∈ Sub(Aut(T )) | σJσ −1 ≤ H (K) for some σ ∈ Aut(T )}
is a neighborhood of H in Sub(Aut(T )). This set only contains locally pro-π groups, so
the conclusion follows.
Remark 5.20. We emphasize that the set of locally pro-π groups in Sub(Aut(T ))0≤C
need not be closed in general. In order to see that, let T be the d-regular tree, with d ≥ 3.
By [BK90, Theorem 7.1 (a)], the group Aut(T ) contains a properly ascending chain of
cocompact lattices Γ1 < Γ2 < . . . . Denoting by
S C the number of Γ1 -orbits of vertices, we
have Γi ∈ Sub(Aut(T ))0≤C for all i. Let H = i≥1 Γi . Since Γ1 is a lattice in H, it follows
that H is unimodular, so that H ∈ Sub(Aut(T ))0≤C . If H were discrete, it would be a
cocompact lattice in Aut(T ), and the chain of inclusions Γ1 ≤ H ≤ Aut(T ) would force
the index [H : Γ1 ] to be finite, contradicting the properly ascending property of the chain
Γ1 < Γ2 < . . . . We infer that H is non-discrete. In particular H is not locally pro-∅. On
the other hand, we have H = limi→∞ Γi by Lemma 2.3 (2). Hence we have constructed a
converging sequence of locally pro-∅ groups in Sub(Aut(T ))0≤C , whose limit is not locally
pro-∅. This confirms that the set of locally pro-π groups in Sub(Aut(T ))0≤C is not closed
in general.
A totally disconnected locally compact group is called locally torsion-free if it has
an open torsion-free subgroup. Typical examples are provided by p-adic analytic groups
(see [DdSMS99, Theorems 4.5 and 8.1]).
We shall need the following basic fact.
Lemma 5.21. Let U be a torsion-free profinite group and U = U0 ≥ U1 ≥ . . . be a
descending chain of open subgroups of U with trivial intersection. Let also p be a positive
integer. For each m ≥ 0, there exists M such that for all u ∈ U , if up ∈ UM then u ∈ Um .
Proof. Suppose the contrary. Then there exist m ≥ 0, a sequence of integers (kn ) tending
to infinity with n, and a sequence (un ) in U such that upn ∈ Ukn and un 6∈ Um . Upon
extracting, we may assume without loss of generality that (un ) converges to some u ∈ U .
Since Um is open and un 6∈ Um for all n, we also have u 6∈ Um . In particular u 6= 1. On
the other hand, we have upn ∈ Ukn , so that up = (limn un )p = limn upn = 1. Hence u is a
non-trivial torsion element of U , a contradiction.
Proposition 5.22. Let T be a locally finite tree all of whose vertices have degree ≥ 2 and
let C > 0. Then the set of locally torsion-free groups is open in Sub(Aut(T ))0≤C .

20

Proof. We can suppose that Sub(Aut(T ))0≤C is non-empty. It follows that T is of bounded
degree. We define the finite set of primes π = {p prime |p ≤ deg(v) ∀v ∈ V (T )} and
observe that the stabilizer Aut(T )v of any vertex v ∈ V (T ) is a pro-π group.
Let H ∈ Sub(Aut(T ))0≤C be locally torsion-free. We must show that H has a neighborhood in Sub(Aut(T ))0≤C that consists of locally torsion-free groups.
[n]

Claim 1. There exist integers M > n > 0 such that for all v ∈ V (T ), p ∈ π and h ∈ Hv ,
[M ]
[n+1]
if hp ∈ Hv then h ∈ Hv
.
Proof of the claim: Since H is locally torsion-free, there exists n0 ≥ 0 and v0 ∈ V (T ) such
[n ]
[n]
that Hv0 0 is torsion-free. As H acts cocompactly on T , there exists n ≥ n0 such that Hv
is torsion-free for all v ∈ V (T ).
Let us now fix a prime p ∈ π and a vertex v ∈ V (T ) and let us apply Lemma 5.21
[n+k]
to the torsion-free profinite groups Uk = Hv
and the integer m = 1. This yields a
[n]
[M (p,v)]
[n+1]
constant M (p, v) ≥ n such that for all h ∈ Hv , if hp ∈ Hv
then h ∈ Hv
.
We next define M as the supremum of M (p, v) taken over all p ∈ π and all vertices v
in a (necessarily finite) fundamental domain for the H-action on V (T ). Then the required
property holds (and we can assume that M > n).

Claim 2. Let M > n be the constants afforded by Claim 1 and let G = H (M ) . Then the
[n]
group Gv is torsion-free for all v ∈ V (T ). In particular G is locally torsion-free.
Proof of the claim: Suppose for a contradiction that for some v ∈ V (T ), there exists a
[n]
non-trivial torsion element in Gv . By the definition of π, every non-trivial torsion element
of Aut(T ) has a power which is a non-trivial element of order p for some p ∈ π. We may
[n]
thus assume that Gv contains a non-trivial element g of prime order p ∈ π. Let then
[k]
k ≥ n be the largest integer such that g ∈ Gv . Then there exists a vertex x ∈ B(v, k)
fixed by g, such that g does not fix B(x, 1) pointwise. It follows that there exists a vertex
[n]
[n+1]
y on the geodesic segment joining v to x such that g ∈ Gy and g 6∈ Gy
.
Since G = H (M ) , there exists h ∈ H such that g|B(y,M ) = h|B(y,M ) . The properties
[n]

[n+1]

[n]

that g ∈ Gy , that gp = 1 and that g 6∈ Gy
respectively imply that h ∈ Hy , that
[M ]
[n+1]
p

. This contradicts Claim 1.
h ∈ Hy and that h 6∈ Hy
From Claim 2 we know that H (M ) is torsion-free. Hence, the set
{J ∈ Sub(Aut(T )) | σJσ −1 ≤ H (M ) for some σ ∈ Aut(T )},
which is a neighborhood of H in Sub(Aut(T )) by Corollary 5.6, only contains locally
torsion-free groups
Remark 5.23. We emphasize that the set of locally torsion-free groups in Sub(Aut(T ))0≤C
need not be closed in general. An excellent illustration of that fact is provided by the main
results of [Stu16], showing that some simple algebraic groups over local fields of positive
characteristic (which are not locally torsion-free) are Chabauty limits of simple algebraic
groups over p-adic fields (which are p-adic analytic, hence locally torsion-free).
21

6

Buildings

6.1

Weyl-transitive automorphism groups of buildings

Let ∆ be a locally finite thick building. A subgroup H of Aut(∆) is said to be Weyltransitive if, for all w ∈ W , the action of H on the ordered pairs (c1 , c2 ) of chambers
such that δ(c1 , c2 ) = w is transitive, where δ : Ch(∆) × Ch(∆) → W is the Weyl-distance.
Remark 6.1. If H ≤ Aut(∆) is strongly transitive on ∆ (i.e. transitive on pairs (A, c)
consisting of an apartment A and a chamber c ∈ A), then it is Weyl-transitive. The
converse holds if ∆ is spherical, but not in general: see [AB08, Proposition 6.14]. If ∆
is of affine type (e.g. ∆ is a tree) and H is closed, it may be seen that if H is Weyltransitive, then it is strongly transitive on the spherical building at infinity of ∆, hence
strongly transitive on ∆ by [CC15, Theorem 1.1]. For ∆ arbitrary (e.g. hyperbolic), the
existence of Weyl-transitive but non-strongly transitive closed subgroups H ≤ Aut(∆) is
likely, but currently we do not know explicit examples.
The following result, which is a straightforward adaptation of [CM11, Corollary 3.1]
dealing with strongly transitive actions, shows that monolithic groups naturally appear
in the context of Weyl-transitive automorphism groups of buildings. It may be seen as a
generalization of Proposition 5.16.
Proposition 6.2. Let ∆ be an infinite irreducible locally finite thick building and H ∈
Sub(Aut(∆)) be Weyl-transitive. Then Mon(H) is topologically simple and transitive on
the set of chambers of ∆. In particular, H/ Mon(H) is compact and Mon(H) = H (∞) .
Proof. We follow the proof of [CM11, Corollary 3.1]. In generalizing from strongly transitive to Weyl-transitive actions, the point requiring a supplementary check is that Tits’
transitivity lemma, which was originally stated for strongly transitive actions, holds more
generally for Weyl-transitive action. This is indeed the case by [AB08, Lemma 3.31]. We
are thus ensured that any non-trivial normal subgroup of H is transitive on the set of
chambers of ∆. Therefore any non-trivial closed normal subgroup of H is cocompact.
Since H is Weyl-transitive on ∆, it is chamber-transitive, hence compactly generated.
We may then invoke [CM11, Theorem E], and conclude the proof word-by-word as in
[CM11, Corollary 3.1]. The argument can be summarized as follows. We know from
[CM11, Theorem E] that the monolith of H is a quasi-product of topologically simple
groups. However, there can be only one simple factor using that the building ∆ has
locally compact CAT(0) metric realization. The desired assertions follow.
The next corollary is then a direct consequence of Theorem 1.3.
Corollary 6.3. Let ∆ be an infinite irreducible locally finite thick building and Γ ≤ Aut(∆)
act cocompactly on ∆. Let Hn → H be a converging sequence in Sub(Aut(∆)) whose limit
H is Weyl-transitive. Suppose that for each n ≥ 1, there exists τn ∈ Aut(∆) such that
τn Γτn−1 ≤ Hn . Then we have
[H : Mon(H)] ≤ lim sup [Hn : Hn(∞) ].
n→∞

22

In particular, if Hn has no proper open subgroup of finite index for each n ≥ 1 then H is
topologically simple.
Proof. This follows from Proposition 6.2 and Theorem 1.3, since a locally finite building
can be seen as a locally finite connected graph whose vertices are the chambers and whose
edges are the pairs of adjacent chambers.
Remark 6.4. If ∆ is a tree, then a closed Weyl-transitive subgroup of Aut(∆) is 2transitive on the set of ends ∂∆. Thus Corollary 5.17 can be deduced from Corollary 6.3.
Remark 6.5. If ∆ is a locally finite Euclidean building of dimension ≥ 2, it can be
seen that there is a unique topologically simple closed subgroup of Aut(∆) acting Weyltransitively: namely the simple algebraic group to which ∆ is associated via Bruhat–Tits
theory. This is of course not the case for trees. For higher-dimensional more exotic buildings (e.g. Bourdon buildings), there can be a much larger collection of simple groups acting
Weyl-transitively, whose variety might potentially be comparable to one encountered in
the case of trees (see [DMSS16]).

6.2

Buildings of virtually free type

We have seen in §5.1 that, for trees, the condition about the common cocompact group Γ
was always fulfilled. It appears that, more generally, it is possible to drop the hypothesis
about Γ in the context of buildings whose associated Coxeter group is virtually free. The
reason is the existence of a strong relation between such buildings and trees.
Lemma 6.6. Let ∆ be an infinite irreducible locally finite thick building of virtually free
type W . Suppose that Aut(∆) is chamber-transitive. Then there exists a locally finite tree
T on which Aut(∆) acts continuously, properly, faithfully and cocompactly.
Proof. By [Dav08, Proposition 8.8.5], W is virtually free if and only if W has a tree of
groups decomposition where each vertex group is a spherical special subgroup. If X is the
tree of groups, then we write X for the underlying tree and denote W = π1 (X ).
Since Aut(∆) is chamber-transitive, we have by [Tit86, Proposition 2] that Aut(∆) =
π1 (X0 ) where X0 has the same underlying tree X as X and has adequate residue stabilizers
as vertex groups and edge groups. By [Ser77, §I.4.5, Théorème 9], we deduce that Aut(∆)
acts on a locally finite tree T containing X so that Aut(∆)\T = X. Moreover, the
stabilizer of a vertex of T in Aut(∆) corresponds to a stabilizer of a spherical residue of ∆
and hence is compact and open. This implies that the action of Aut(∆) on T is continuous
and proper. Finally, the kernel K ≤ Aut(∆) of this action on T stabilizes all residues of ∆
of a fixed spherical type. Since ∆ is infinite and irreducible, this implies that K is trivial
(see [AB10, Main Theorem]). The action is thus faithful.
Remark 6.7. The tree of group decomposition of W is generally not unique. In particular,
the tree T and the Aut(∆)-action on T afforded by Lemma 6.6 are not canonical.

23

Corollary 6.8. Let ∆ be an infinite irreducible locally finite thick building of virtually
free type W . Let Hn → H be a converging sequence in Sub(Aut(∆)) whose limit H is
Weyl-transitive. Then we have
[H : Mon(H)] ≤ lim sup [Hn : Hn(∞) ].
n→∞

In particular, if Hn has no proper open subgroup of finite index for each n ≥ 1 then H
is topologically simple.
Proof. Let T be the locally finite tree given by Lemma 6.6. The fact that Aut(∆) acts
continuously, properly and faithfully on T means that there is a map i : Aut(∆) → Aut(T )
which is an isomorphism onto its image, the latter being closed in Aut(T ). We thus have
the converging sequence i(Hn ) → i(H) in Sub(Aut(T )), such that i(H) acts cocompactly
on T and is unimodular (as an amalgamated sum of compact subgroups). The conclusion
then follows from Corollary 5.7, since Mon(H) = H (∞) (see Proposition 6.2).

A

The clopen subset STAlt ⊆ ST

Let T be a thick locally finite tree. As before, we denote by ST the set of isomorphism
classes of groups in Sub(Aut(T )) which are topologically simple and 2-transitive on ∂T .
By Theorem 1.2, this set carries a compact Hausdorff topology induced from the Chabauty
topology on Sub(Aut(T )). The goal of this appendix is to provide supplementary information on that compact space.
First note that, when H ∈ Sub(Aut(T )) is 2-transitive on ∂T , the action of the
stabilizer Hv of a vertex v ∈ V (T ) in H is 2-transitive on the set of neighbors of v (see
[BM00, Lemma 3.1.1]). In particular, H must be edge-transitive, so that ST is empty
when T is not semi-regular.
On the contrary, if T is semi-regular then ST contains the isomorphism class of Aut(T )+
(which is simple by [Tit70]). In [Rad15], the second-named author restricted his attention
to the groups H which locally contain the full alternating group, i.e. such that the action
of Hv on its set of d neighbors contains Alt(d) for each v ∈ V (T ). Let us denote by STAlt
the subset of ST consisting of the isomorphism classes of all those groups. An exhaustive
description of the set STAlt when the vertices of T have degree ≥ 6 is given in [Rad15].
Below we summarize some of its properties.
Proposition A.1. Let T be the (d0 , d1 )-semi-regular tree. Then STAlt is a closed open
subset of ST containing the isomorphism class of Aut(T )+ .
Moreover, if d0 , d1 ≥ 6, then the compact space STAlt is countably infinite and its second
Cantor–Bendixson derivative is {[Aut(T )+ ]}.
Proof. The first assertion is clear. In order to prove the second one, we freely use the
terminology and notation from [Rad15] without repeating all the definitions in full details.
In that paper, a legal coloring i of T is fixed and, given two possibly empty finite subsets
Y0 , Y1 ⊂ N, a group G+
(i) (Y0 , Y1 ) is defined. Let us describe their properties which will
24

be needed here. We assume henceforth that d0 , d1 ≥ 6. The group G+
(i) (∅, ∅) is exactly

Aut(T )+ , while G+
(i) ({0}, {0}) is the semiregular analog of the universal locally alternating
+
+
group of Burger–Mozes [BM00]. We call it U(i)
(Alt) ≤
(Alt). For all Y0 and Y1 we have U(i)
+
+
+
G(i) (Y0 , Y1 ) ≤ Aut(T ) . The groups G(i) (Y0 , Y1 ) locally contain the alternating group
+
(Alt) which does), are boundary-2-transitive and abstractly simple
(since they contain U(i)
(see [Rad15, Theorem A (i), (ii)]) and, up to conjugation, these are the only such groups
(see [Rad15, Theorem B (ii)]). If [G] denotes the isomorphism class of G ∈ Sub(Aut(T )),
then this means that
STAlt = {[G+
(i) (Y0 , Y1 )] | Y0 , Y1 ⊂ N are finite}.
In fact, the groups G+
(i) (Y0 , Y1 ) are not pairwise distinct, but this is not important for
the following discussion. We now give some other properties of these groups. In order to
shorten the statements, we adopt the convention max(∅) := +∞.
Fact 1. If (X (n) ) is a sequence of finite subsets of N such that max X (n) → +∞ and if Y
(n) , Y ) → G+ (∅, Y ) and G+ (Y, X (n) ) → G+ (Y, ∅).
is a finite subset of N, then G+
(i)
(i)
(i)
(i) (X
Proof of the fact: When X is a non-empty finite subset of N and Y is a finite subset of
+
+
(max X) (see [Rad15, §4.1]). If (X (n) ) is a
N, we have G+
(i) (X, Y ) ≤ G(i) (∅, Y ) ≤ G(i) (X, Y )
(n) , Y ) → G+ (∅, Y )
sequence such that max X (n) → +∞, then we can deduce that G+
(i)
(i) (X
with Lemma 2.1. Indeed, (i) is clear and (ii) can be obtained as follows. Fix h ∈ G+
(i) (∅, Y )
+
(n) , Y )(max X
and v0 ∈ V (T ). For each n ≥ 1, since G+
(i) (∅, Y ) ≤ G(i) (X

(n) )

(we can assume

(n) , Y ) such that h|
that
is non-empty), there exists hn ∈ G+
B(v0 ,max X (n) ) =
(i) (X
(n)
hn |B(v0 ,max X (n) ) . Then hn → h because max X → +∞, which proves (ii). The reasoning
(n) ) → G+ (Y, ∅).
is exactly the same to obtain that G+

(i) (Y, X
(i)

X (n)

−1 ≤ G+ (Y , Y ) for some finite subsets Y , Y , Z , Z ⊂ N and
Fact 2. If σG+
0 1
0
1
(i) (Z0 , Z1 )σ
(i) 0 1
+
+
+
+
some σ ∈ Aut(T ), then either G(i) (Z0 , Z1 ) ≤ G(i) (Y0 , Y1 ) or G(i) (Z1 , Z0 ) ≤ G(i) (Y0 , Y1 )
+
Proof of the fact: If σ ∈ Aut(T )+ then G+
(i) (Z0 , Z1 ) ≤ G(i) (Y0 , Y1 ) by [Rad15, Lemma 4.10
(i)]. If σ ∈ Aut(T )\Aut(T )+ , then there exists a particular element ν ∈ Aut(T )\Aut(T )+
−1 = G+ (Z , Z ) and the conclusion follows.

such that νG+
1
0
(i)
(i) (Z0 , Z1 )ν
+
Fact 3. If G+
(i) (Z0 , Z1 ) ≤ G(i) (Y0 , Y1 ) for some finite subsets Y0 , Y1 , Z0 , Z1 ⊂ N, then
max Z0 ≤ max Y0 and max Z1 ≤ max Y1 .
′ (0), K ′ (1) ∈ N ∪{+∞} are asProof of the fact: In [Rad15, §5.3], two invariants KH
H
+
(Alt). These invariants
sociated to any closed subgroup H ≤ Aut(T ) containing U(i)
′ (1) ≤ K ′ (1). For
′
′
′
have the property that if H ≤ H then KH (0) ≤ KH ′ (0) and KH
H′
+
′
′
H = G(i) (Y0 , Y1 ), we have KH (0) = max Y0 and KH (1) = max Y1 (see [Rad15, Table 1]),
which suffices to conclude.


25

Fact 4. For all finite subsets Y0 , Y1 ⊂ N, there exists an integer K ≥ 0 such that
(K) = G+ (Y , Y ).
G+
(i) 0 1
(i) (Y0 , Y1 )
Proof of the fact: See [Rad15, Theorem H].



Let us now compute the Cantor-Bendixson derivatives of STAlt . In STAlt , the points
+
+
and [G+
(i) (∅, Y )] are not isolated. Indeed, [G(i) (Y, {n})] → [G(i) (Y, ∅)] when
+
n → +∞ by Fact 1, and [G+
(i) (Y, {n})] 6= [G(i) (Y, ∅)] for each n ≥ 0 (Facts 2 and 3). We
claim that the points [G+
(i) (Y0 , Y1 )] with Y0 and Y1 non-empty are isolated. Suppose for a

[G+
(i) (Y, ∅)]

(n)

(n)

contradiction that there exists sequences (Y0 ), (Y1 ) and (τn ) with τn ∈ Aut(T ) such
(n)
(n)
(n)
(n) −1
+
+
+
that τn G+
(i) (Y0 , Y1 )τn → G(i) (Y0 , Y1 ) and with [G(i) (Y0 , Y1 )] 6= [G(i) (Y0 , Y1 )] for
each n ≥ 1. Then, by Fact 4 and Corollary 5.6, there exists (σn ) with σn ∈ Aut(T ) such
(n)
(n) −1
+
that σn G+
(i) (Y0 , Y1 )σn ≤ G(i) (Y0 , Y1 ) for sufficiently large n. From Fact 2, we deduce
(n)

(n)

(n)

(n)

(Y , Y1 ) ≤ G+
(Y , Y0 ) ≤ G+
that G+
(Y , Y ) or G+
(Y , Y ) for all sufficiently
(i) 0
(i) 1
(i) 0 1
(i) 0 1

+
large n. But there are only finitely many Z0 , Z1 ⊂ N such that G+
(i) (Z0 , Z1 ) ≤ G(i) (Y0 , Y1 )
(see Fact 3), so we cannot have the supposed convergence. We just proved that the first
Cantor-Bendixson derivative of STAlt is

(∅, Y )] | Y ⊂ N is finite}.
(Y, ∅)], [G+
(STAlt )′ = {[G+
(i)
(i)
With the exact same reasoning, we then obtain that
+
(STAlt )′′ = {[G+
(i) (∅, ∅)]} = {[Aut(T ) ]}.

Remark A.2. Proposition A.1 implies (see [MS20, Théorème 1]) that STAlt is homeomor2

phic to the space N̂ , where N̂ is the one-point compactification of the discrete space N
(which is homeomorphic to the compact subset 1, 21 , 13 , . . . , 0 of the real line).
Remark A.3. If the degrees d0 and d1 of the vertices of T are such that each 2-transitive
subgroup of Sym(dt ) contains Alt(dt ) (for each t ∈ {0, 1}), then ST = STAlt . For example,
this is the case for d0 = d1 = 3. If in addition d0 , d1 ≥ 6, then the classification theorem
from [Rad15] applies, and therefore yields a complete description of ST . One should note
that the set of natural numbers d such that each 2-transitive subgroup of Sym(d) contains
Alt(d) is asymptotically dense in N (see [Rad15, Corollary B.2]).
Remark A.4. It is actually a direct consequence of [Rad15, Theorem A (i), (ii) and
Theorem B (i)] that the space STAlt (and hence also ST ) is infinite when d0 , d1 ≥ 4. The
case where d0 = 3 or d1 = 3 is not explicitly dealt with in [Rad15], but one can show
that STAlt is infinite also in that case. Indeed, the definition of the groups G+
(i) (Y0 , Y1 )
(where Y0 , Y1 are finite subsets of N) from loc. cit. makes sense for all d0 , d1 ≥ 3. For
these groups to be boundary-2-transitive, one however needs to require Y0 6= {0} (resp.
Y1 6= {0}) when d0 = 3 (resp. d1 = 3). Under the latter hypothesis, it is then possible to
adapt the ideas from [Rad15, §4] and show that these groups are abstractly simple and that
they represent infinitely many isomorphism classes. In the specific case of the trivalent
26

tree T3 , the infiniteness of ST3 can alternatively be established using rank one simple
algebraic groups over local fields with residue field of order 2. An exhaustive description
of the subset of ST3 consisting of (isomorphism classes of) algebraic groups may be found
in [Stu16].

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