to describe combinatorially the asymptotically rigid mapping class groups studied in [38], including the braided Ptolemy–Thompson groups from [29] and the braided Houghton groups from [21, 28].

**5.3. Monoid presentations**

So far, we have only considered diagram groups defined from semigroup presenta-
tions, but it is also possible to start from monoid presentations, i.e., allowing relations
of the form $u = 1$, where 1 corresponds to the empty word. Given a monoid present-
ation $\mathcal{P} = (\Sigma|\mathcal{R})$, one can define a Squier square complex $S(\mathcal{P})$ by repeating word
for word Definition 2.1; and, fixing a baseword $w \in \Sigma^+$, the diagram group $D(\mathcal{P}, w)$
is defined as the fundamental group of the connected component $S(\mathcal{P}, w)$ of $S(\mathcal{P})$.
Following Section 2.2, a diagrammatic description of $D(\mathcal{P}, w)$ can be obtained, but
it is more convenient to use pictures instead of diagrams. The main difference with
semigroup presentations is that, in a picture over our monoid presentation, a transistor
labelled by a relation $u = 1$ (resp., $1 = u$) has not bottom (resp., top) wires.

<figure><img src="image_39.png" /></figure>

Two examples of dipole reduction.

A reduced picture.

We refer to [45] for more details. Even though many techniques can be transferred
from diagram groups over semigroup presentation to diagram groups over monoid
presentations, a major difference is that a picture over monoid presentation may admit
several reduced forms, i.e., reducing the dipoles in one order or another may lead to
distinct pictures. See Figure 20.

Even worse, the problem of determining whether or not two given pictures are
equivalent modulo dipole reduction may be undecidable [45, Theorem 13.3]. This
explains why it is usually much more difficult to work with diagram groups over
monoid presentations than diagram groups over semigroup presentations. As an illus-
tration of this phenomenon, let us mention the following open question.

**Question 5.10 ([45])**. Are diagram groups over monoid presentations torsion-free?
<figure><img src="image_40.png" /><figcaption>**Figure 20.** Two distinct reductions of a picture over $\langle x | x = 1 \rangle$.</figcaption></figure>

This question illustrates how little is known about diagram groups over monoid presentations.

It is worth noticing that Question 5.10 cannot be answered with the help of median geometry as mentioned in Section 4.2. Indeed, a (connected component of the) Squier cube complex over a monoid presentation may not be nonpositively curved. For instance, if $\mathcal{P} = \langle x | x = 1 \rangle$, then the 4-cycle

$$xx \xrightarrow{(x,x=1,1)} x \xrightarrow{(1,x=1,1)} 1 \xrightarrow{(1,1=x,1)} x \xrightarrow{(1,x=1,x)} xx$$

shows that the link of the vertex 1 is not simplicial since there is a loop based at the vertex given by the edge $(1, 1 = x, 1)$.

# 6. Open questions

In this final section, we record various open questions related to diagram groups.

## Classification of diagram groups

In view of the known examples of diagram groups, there seem to be three main fami-
ilies of diagram groups: those which contain a copy of Thompson's group $F$, those
which contain a copy of the wreath product $Z \rtimes Z$ but not of $F$, and those which embed
into a right-angled Artin group. Are there other big families of diagram groups? Let
us ask a few precise questions in this direction.

**Conjecture 6.1 ([33]).** A diagram group embeds into a right-angled Artin group if and only if it does not contain $Z \rtimes Z$.

**Question 6.2 ([33]).** Is a diagram group not containing Thompson's group $F$ linear or residually finite?
It is worth noticing that, given a semigroup presentation $\mathcal{P} = \langle \Sigma | \mathcal{R} \rangle$ and a base-word $w \in \Sigma^+$, we know exactly when the diagram group $D(\mathcal{P}, w)$ contains $F$ or $\mathbb{Z} \not\cong \mathbb{Z}$. Indeed, $D(\mathcal{P}, w)$ contains $F$ if and only if there exist words $a, b, x$ such that $w = axb$ and $x^2 = x$ hold modulo $\mathcal{P}$ [48]; and $D(\mathcal{P}, w)$ contains $\mathbb{Z} \not\cong \mathbb{Z}$ if and only if there exist words $a, b, p$ such that $D(\mathcal{P}, p) \neq \{1\}$ and such that $w = ab, a = ap, b = pb$ hold modulo $\mathcal{P}$ [47].

**Question 6.3 ([50]).** If a diagram group does not contain any non-abelian free subgroup, does it embed into Thompson's group $F$?

Notice that Theorem 4.46 implies that a diagram with no free non-abelian subgroup maps to $F$ with an abelian kernel.

## Isomorphism problem

Given any reasonable family of groups, a basic, but usually difficult, problem is to find how to determine whether two members of the family yield isomorphic groups.

**Question 6.4.** Given two semigroup presentations $\mathcal{P}_1 = \langle \Sigma_1 | \mathcal{R}_1 \rangle$, $\mathcal{P}_2 = \langle \Sigma_2 | \mathcal{R}_2 \rangle$ and two basewords $w_1 \in \Sigma_1^+$, $w_2 \in \Sigma_2^+$, how to determine whether the diagram groups $D(\mathcal{P}_1, w_1)$ and $D(\mathcal{P}_2, w_2)$ are isomorphic?

As an easy sufficient condition, notice that, given a semigroup presentation $\mathcal{P} = \langle \Sigma | \mathcal{R} \rangle$ and two words $u, v \in \Sigma^+$, the diagram groups $D(\mathcal{P}, u)$ and $D(\mathcal{P}, v)$ are isomorphic if $u$ and $v$ represent the same element in the semigroup defined by $\mathcal{P}$. This is an immediate consequence of the fact that $u$ and $v$ are two vertices in the same connected component of the Squier square complex $S(\mathcal{P})$, which implies that the fundamental groups are conjugate in the fundamental groupoid.

More interesting conditions are given by [49, Theorem 4.1]. Nevertheless, we are far from a reasonable answer to Question 6.4. In order to illustrate the difficulty of the problem, let us mention a few examples of non-trivial isomorphisms. Considering the semigroup presentations

$$ \begin{align*} \mathcal{A} &= \langle a, b, c \mid a = b, b = c, c = a \rangle, \\ \mathcal{B} &= \langle a, b, p \mid a = ap, b = pb \rangle, \\ \mathcal{C} &= \langle a, b, x, y \mid ax = ay, xb = yb \rangle, \end{align*} $$

the diagram groups $D(\mathcal{A}, a)$, $D(\mathcal{B}, ab)$, and $D(\mathcal{C}, axb)$ are all infinite cyclic. Considering the semigroup presentations

$$ \begin{align*} \mathcal{D} &= \langle a, b, c, p \mid a = b, b = c, c = a, a = ap \rangle, \\ \mathcal{E} &= \langle a, b, c, x, y, p \mid a = b, b = c, c = a, ax = ay, xp = yp \rangle, \\ \mathcal{F} &= \langle a, b, c \mid ab = ba, ac = ca, bc = cb \rangle, \end{align*} $$
the diagram groups $D(\mathcal{D}, a)$, $D(\mathcal{E}, axp)$, and $D(\mathcal{F}, abc^2)$ are all free of rank two.
Considering the presentation

$$
\begin{align*}
\mathcal{G} &= \langle a, b, c \mid a = bc, b = ca, c = ab \rangle, \\
\mathcal{H} &= \langle x \mid x = x^9 \rangle,
\end{align*}
$$

the diagram groups $D(\mathcal{G}, a)$ and $D(\mathcal{H}, x)$ are both isomorphic to the generalised
Thompson group $F_9$ [46].

The triviality problem is probably a more reasonable question before attacking
Question 6.4.

**Question 6.5.** Given a semigroup presentation $\mathcal{P} = \langle \Sigma | \mathcal{R} \rangle$ and a baseword $w \in \Sigma^+$, how to determine whether the diagram group $D(\mathcal{P}, w)$ is non-trivial?

Algorithmic aspects

In addition to the isomorphism and triviality problems previously mentioned, there
are other interesting problems that remain to be solved algorithmically. For instance,
consider the following question.

**Question 6.6 ([45]).** Given a finite semigroup presentation $\mathcal{P} = \langle \Sigma | \mathcal{R} \rangle$ and a base-word $w \in \Sigma^+$, is there an algorithm computing the rank of the abelianisation of the diagram group $D(\mathcal{P}, w)$?

In free groups, many problems are solved algorithmically thanks to Stallings' fold-
ings. As mentioned in Section 3.5, a similar theory exists for directed 2-complexes and
their diagram groups. However, the theory remains to be developed.

**Problem 6.7.** Develop the theory of foldings of directed 2-complexes for diagram groups. In particular, solve [41, Conjecture 3.11].

Nevertheless, it is worth mentioning that Stallings’s foldings have been applied
with success to Thompson’s group $F$; see, for instance, [41–43].

Solvable diagram groups

Theorem 2.23 shows that there are no interesting nilpotent diagram groups. There
are interesting solvable diagram groups, such as the wreath product $Z \ltimes Z$, but only
few examples are known. Finding more examples, or proving that there are only few
possible examples, would be interesting.

**Question 6.8.** Which metabelian (resp., solvable) groups are diagram groups?

**Question 6.9.** Are there finitely presented solvable diagram groups?

**Conjecture 6.10.** A diagram group does not contain $Z \ltimes Z^2$.
## Right-angled Artin groups

As mentioned in Section 2.4, right-angled Artin groups given by finite trees and finite interval graphs are diagram groups. On the other hand, we saw with Corollary 3.19 that right-angled Artin groups defined by triangle-free graphs with induced cycles of odd length $\ge 5$ are not diagram groups. This naturally raises the following question.

**Question 6.11.** Which (finitely generated) right-angled Artin groups can be described as diagram groups?

So far, neither an expected nor a reasonable answer is known. A related question is the following.

**Question 6.12 ([33]).** Are all right-angled Artin groups subgroups of diagram groups?

For instance, even though right-angled Artin groups defined by cycles of odd lengths $\ge 5$ are not diagram groups, they do embed into diagram groups<fnref n="4" /> [33].

## Miscellaneous

Recall that, given two groups $G, H$, the group $G$ is *residually* $H$ if, for every non-trivial $g \in G$, there exists a morphism $\varphi : G \to H$ satisfying $\varphi(g) \neq 1$.

**Question 6.13 ([45, 50]).** Are all diagram groups residually $\text{PL}_+([0, 1])$? or even residually $F$?

A possible strategy, suggested in [45], was the following. Let $\mathcal{P} = \langle \Sigma | \mathcal{R} \rangle$ be a semigroup presentation and let $w \in \Sigma^+$ be a baseword. Every atomic $(aub, avb)$-diagram $\Delta$ over $\mathcal{P}$ can be written as $\varepsilon(a) + \Psi(u = v) + \varepsilon(b)$, where $\varepsilon(a), \varepsilon(b)$ are diagrams with no cells labelled by $a, b$ and where $\Psi(u = v)$ is a single cell labelled by $u = v$. To such a diagram $\Delta$, we associate a continuous, nondecreasing, piecewise linear homeomorphism $H(\Delta) : [0, |aub|] \to [0, |avb|]$ which has slope 1 on both $[0, |a|]$ and $[|au|, |aub|]$. Then, one gets a morphism

$$ \rho : D(\mathcal{P}, w) \to \text{PL}_+([0, |w|]) $$

by taking a diagram $\Phi$, decomposing it as a concatenation of atomic diagrams $\Phi_1 \circ \cdots \circ \Phi_n$, and sending it to $H(\Phi_1) \circ \cdots \circ H(\Phi_n)$. Is it possible to obtain a non-trivial homeomorphism $\rho(\Phi)$ by choosing the homeomorphisms $H(\cdot)$ carefully?

<footnote n="4">⁴This is contrary to what [19] claims. There, the authors cite [47] in order to justify that such right-angled Artin groups do not embed into diagram groups. But the result only proves that these right-angled Artin groups are not diagram groups themselves, and the proof (sketched in Section 3.3) does not imply this statement either.</footnote>
We saw a particular instance of this construction in Section 2.4, where we described how to embed $D(\langle x | x = x^2 \rangle, x)$ into $PL_+([0, 1])$ and recognised that it coincides with Thompson’s group $F$. However, it turns out that the strategy described above cannot work in full generality, as shown by [50, Example 7.7]. Nevertheless, the construction remains interesting and still applies to many examples.

As mentioned in Section 2.5, it is proved in [32] that diagram groups with $\mathbb{Z}^2$ are locally free, which implies in particular that there are no interesting hyperbolic diagram groups. First, this raises the following question.

**Question 6.14 ([32])**. Is a locally free diagram group necessarily free?

Next, since we saw in Section 4.6 that there are many interesting examples of acylindrically hyperbolic diagram groups, it is natural to look for relatively hyperbolic diagram groups. We conjecture that there are no interesting examples there either.

**Conjecture 6.15.** Relatively hyperbolic diagram groups are free products.

**Question 6.16 ([49])**. Do the cohomological and algebraic dimensions of diagram groups always coincide?

Recall that the algebraic dimension of a group is the maximal rank of a free abelian subgroup. According to [50], the answer is positive for diagram groups over complete presentations.

**Question 6.17 ([47])**. Is every subgroup of Thompson’s group $F$ a diagram group?

**Question 6.18 ([47])**. Is the derived subgroup of a diagram group again a diagram group?

Recall from Section 4.4 that a finitely generated diagram group satisfies the Burillo property if every word length is bi-Lipschitz equivalent to the diagram length $\#(\cdot)$, i.e., the number of 2-cells in a reduced representative.

**Question 6.19 ([2])**. Does there exist a finitely generated diagram that does not satisfy the Burillo property?

There are examples of infinitely presented diagram groups (such as abelian and non-abelian free groups of infinite rank), examples of finitely generated diagram groups that are not finitely presented (such as $\mathbb{Z} \ltimes \mathbb{Z}$ and $\mathbb{Z} \bullet \mathbb{Z}$), examples of diagram groups of type $F_\infty$ but not of type $F$ (such as Thompson’s group $F$), and examples of diagram groups of type $F$ (such as abelian and non-abelian free groups). But there is a gap in the progression; hence, the following question arises.

**Question 6.20.** Given an $n \ge 2$, does there exist a diagram group of type $F_n$ but not $F_{n+1}$?
It is worth noticing that such examples exist for symmetric diagram groups: the Houghton group $H_n$ is of type $F_{n-1}$ but not of type $F_n$ [12].

**Acknowledgements.** I am grateful to the anonymous referee for their comments on the manuscript, and to Mateo Tarocchi for having pointed out to me a mistake in the proof of Proposition 2.14 from the first version of the article.

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Received 9 December 2022.

**Anthony Genevois**

Institut Mathématiques Alexander Grothendieck, University of Montpellier, Place Eugène Bataillon, 34090 Montpellier, France; anthony.genevois@umontpellier.fr