Motivated by approaches to the word problem for one-relation monoids arising from work of Adian and Oganesian (1987), Guba (1997), and Ivanov, Margolis, and Meakin (2001), we study the submonoid and rational subset membership problems in one-relation monoids and in positive one-relator groups. We give the first known examples of positive one-relator groups with undecidable submonoid membership problem, and we apply this to give the first known examples of one-relation monoids with undecidable submonoid membership problem. We construct several infinite families of one-relation monoids with undecidable submonoid membership problem, including examples that are defined by relations of the form $w=1$ but which are not groups, and examples defined by relations of the form $u=v$ where both of u and v are nonempty. As a consequence, we obtain a classification of the right-angled Artin groups that can arise as subgroups of one-relation monoids. We also give examples of monoids with a single defining relation of the form $aUb = a$ and examples of the form $aUb=aVa$, with undecidable rational subset membership problem. We give a one-relator group defined by a freely reduced word of the form $uv^{-1}$ with $u, v$ positive words, in which the prefix membership problem is undecidable. Finally, we prove the existence of a special two-relator inverse monoid with undecidable word problem, and in which both the relators are positive words. As a corollary, we also find a positive two-relator group with undecidable prefix membership problem. In proving these results, we introduce new methods for proving undecidability of the rational subset membership problem in monoids and groups, including by finding suitable embeddings of certain trace monoids.

A nontrivial element of a group is a generalized torsion element if some products of its conjugates is the identity. The minimum number of such conjugates is called a generalized torsion order. We provide several restrictions for generalized torsion orders by using the Alexander polynomial.

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$, where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.

We introduce two families of two-generator one-relator groups called primitive extension groups and show that a one-relator group is hyperbolic if its primitive extension subgroups are hyperbolic. This reduces the problem of characterizing hyperbolic one-relator groups to characterizing hyperbolic primitive extension groups. These new groups, moreover, admit explicit decompositions as graphs of free groups with adjoined roots. In order to obtain this result, we characterize $2$-free one-relator groups with exceptional intersection in terms of Christoffel words, show that hyperbolic one-relator groups have quasi-convex Magnus subgroup, and build upon the one-relator tower machinery developed in previous work of the author.

The complete classification of the finite simple groups that are $(2,3)$-generated is a problem which is still open only for orthogonal groups. Here, we construct $(2, 3)$-generators for the finite odd-dimensional orthogonal groups $\Omega _{2k+1}(q)$, $k\geq 4$. As a byproduct, we also obtain $(2,3)$-generators for $\Omega _{4k}^+(q)$ with $k\geq 3$ and q odd, and for $\Omega _{4k+2}^\pm (q)$ with $k\geq 4$ and $q\equiv \pm 1~ \mathrm {(mod~ 4)}$.

We investigate the groups of units of one-relator and special inverse monoids. These are inverse monoids which are defined by presentations, where all the defining relations are of the form $r=1$. We develop new approaches for finding presentations for the group of units of a special inverse monoid, and apply these methods to give conditions under which the group admits a presentation with the same number of defining relations as the monoid. In particular, our results give sufficient conditions for the group of units of a one-relator inverse monoid to be a one-relator group. When these conditions are satisfied, these results give inverse semigroup theoretic analogues of classical results of Adjan for one-relator monoids, and Makanin for special monoids. In contrast, we show that in general these classical results do not hold for one-relator and special inverse monoids. In particular, we show that there exists a one-relator special inverse monoid whose group of units is not a one-relator group (with respect to any generating set), and we show that there exists a finitely presented special inverse monoid whose group of units is not finitely presented.

A prefix monoid is a finitely generated submonoid of a finitely presented group generated by the prefixes of its defining relators. Important results of Guba (1997), and of Ivanov, Margolis and Meakin (2001), show how the word problem for certain one-relator monoids, and inverse monoids, can be reduced to solving the membership problem in prefix monoids of certain one-relator groups. Motivated by this, in this paper, we study the class of prefix monoids of finitely presented groups. We obtain a complete description of this class of monoids. All monoids in this family are finitely generated, recursively presented and group-embeddable. Our results show that not every finitely generated recursively presented group-embeddable monoid is a prefix monoid, but for every such monoid, if we take a free product with a suitably chosen free monoid of finite rank, then we do obtain a prefix monoid. Conversely, we prove that every prefix monoid arises in this way. Also, we show that the groups that arise as groups of units of prefix monoids are precisely the finitely generated recursively presented groups, whereas the groups that arise as Schützenberger groups of prefix monoids are exactly the recursively enumerable subgroups of finitely presented groups. We obtain an analogous result classifying the Schützenberger groups of monoids of right units of special inverse monoids. We also give some examples of right cancellative monoids arising as monoids of right units of finitely presented special inverse monoids, and we show that not all right cancellative recursively presented monoids belong to this class.

In this paper, we study intersection configurations – which describe the behaviour of multiple (finite) intersections of subgroups with respect to finite generability – in the realm of free and free times free-abelian (FTFA) groups. We say that a configuration is realizable in a group $G$ if there exist subgroups $H_1,\ldots, H_k \leqslant G$ realizing it. It is well known that free groups ${\mathbb {F}_{n}}$ satisfy the Howson property: the intersection of any two finitely generated subgroups is again finitely generated. We show that the Howson property is indeed the only obstruction for multiple intersection configurations to be realizable within nonabelian free groups. On the contrary, FTFA groups ${\mathbb {F}_{n}} \times \mathbb {Z}^m$ are well known to be non-Howson. We also study multiple intersections within FTFA groups, providing an algorithm to decide, given $k\geq 2$ finitely generated subgroups, whether their intersection is again finitely generated and, in the affirmative case, compute a ‘basis’ for it. We finally prove that any intersection configuration is realizable in an FTFA group ${\mathbb {F}_{n}} \times \mathbb {Z}^m$, for $n\geq 2$ and large enough $m$. As a consequence, we exhibit finitely presented groups where every intersection configuration is realizable.

Consider the following classes of pairs consisting of a group and a finite collection of subgroups:

• • $ \mathcal{C}= \left \{ (G,\mathcal{H}) \mid \text{$\mathcal{H}$ is hyperbolically embedded in $G$} \right \}$

• • $ \mathcal{D}= \left \{ (G,\mathcal{H}) \mid \text{the relative Dehn function of $(G,\mathcal{H})$ is well-defined} \right \} .$

Let $G$ be a group that splits as a finite graph of groups such that each vertex group $G_v$ is assigned a finite collection of subgroups $\mathcal{H}_v$, and each edge group $G_e$ is conjugate to a subgroup of some $H\in \mathcal{H}_v$ if $e$ is adjacent to $v$. Then there is a finite collection of subgroups $\mathcal{H}$ of $G$ such that

1. 1. If each $(G_v, \mathcal{H}_v)$ is in $\mathcal C$, then $(G,\mathcal{H})$ is in $\mathcal C$.

2. 2. If each $(G_v, \mathcal{H}_v)$ is in $\mathcal D$, then $(G,\mathcal{H})$ is in $\mathcal D$.

3. 3. For any vertex $v$ and for any $g\in G_v$, the element $g$ is conjugate to an element in some $Q\in \mathcal{H}_v$ if and only if $g$ is conjugate to an element in some $H\in \mathcal{H}$.

That edge groups are not assumed to be finitely generated and that they do not necessarily belong to a peripheral collection of subgroups of an adjacent vertex are the main differences between this work and previous results in the literature. The method of proof provides lower and upper bounds of the relative Dehn functions in terms of the relative Dehn functions of the vertex groups. These bounds generalize and improve analogous results in the literature.

A generating set for a finite group G is minimal if no proper subset generates G, and $m(G)$ denotes the maximal size of a minimal generating set for G. We prove a conjecture of Lucchini, Moscatiello and Spiga by showing that there exist $a,b> 0$ such that any finite group G satisfies $m(G) \leqslant a \cdot \delta (G)^b$, for $\delta (G) = \sum _{p \text { prime}} m(G_p)$, where $G_p$ is a Sylow p-subgroup of G. To do this, we first bound $m(G)$ for all almost simple groups of Lie type (until now, no nontrivial bounds were known except for groups of rank $1$ or $2$). In particular, we prove that there exist $a,b> 0$ such that any finite simple group G of Lie type of rank r over the field $\mathbb {F}_{p^f}$ satisfies $r + \omega (f) \leqslant m(G) \leqslant a(r + \omega (f))^b$, where $\omega (f)$ denotes the number of distinct prime divisors of f. In the process, we confirm a conjecture of Gill and Liebeck that there exist $a,b> 0$ such that a minimal base for a faithful primitive action of an almost simple group of Lie type of rank r over $\mathbb {F}_{p^f}$ has size at most $ar^b + \omega (f)$.

A. Mark and J. Paupert [Presentations for cusped arithmetic hyperbolic lattices, 2018, arXiv:1709.06691.] presented a method to compute a presentation for any cusped complex hyperbolic lattice. In this note, we will use their method to give a presentation for the Eisenstein-Picard modular group in three complex dimensions.

Let G be a Baumslag–Solitar group. We calculate the intersection $\gamma_{\omega}(G)$ of all terms of the lower central series of G. Using this, we show that $[\gamma_{\omega}(G),G]=\gamma_{\omega}(G)$, thus answering a question of Bardakov and Neschadim [1]. For any $c \in \mathbb{N}$, with $c \geq 2$, we show, by using Lie algebra methods, that the quotient group $\gamma_{c}(G)/\gamma_{c+1}(G)$ of the lower central series of G is finite.

A generating set S for a group G is independent if the subgroup generated by $S\setminus \{s\}$ is properly contained in G for all $s \in S.$ We describe the structure of finite groups G such that there are precisely two numbers appearing as the cardinalities of independent generating sets for G.

We study the following decision problem: given an exponential equation $a_1g_1^{x_1}a_2g_2^{x_2}\dots a_ng_n^{x_n}=1$ over a recursively presented group G, decide if it has a solution with all $x_i$ in $\mathbb {Z}$. We construct a finitely presented group G where this problem is decidable for equations with one variable and is undecidable for equations with two variables. We also study functions estimating possible solutions of such an equation through the lengths of its coefficients with respect to a given generating set of G. Another result concerns Turing degrees of some natural fragments of the above problem.

The higher-dimensional Thompson groups $nV$, for $n \geqslant 2$, were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra 284 (2005), 520–558]. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in $nV$ and reflecting the self-similar structure of n-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.

How many 2-cells must two finite CW-complexes have to admit a common, but not finite common, covering? Leighton’s theorem says that both complexes must have 2-cells. We construct an almost (?) minimal example with two 2-cells in each complex.

In this paper, we consider the $T$- and $V$-versions, ${T_\tau }$ and ${V_\tau }$, of the irrational slope Thompson group ${F_\tau }$ considered in J. Burillo, B. Nucinkis and L. Reeves [An irrational-slope Thompson's group, Publ. Mat. 65 (2021), 809–839]. We give infinite presentations for these groups and show how they can be represented by tree-pair diagrams similar to those for $T$ and $V$. We also show that ${T_\tau }$ and ${V_\tau }$ have index-$2$ normal subgroups, unlike their original Thompson counterparts $T$ and $V$. These index-$2$ subgroups are shown to be simple.

A group is $\frac 32$-generated if every non-trivial element is part of a generating pair. In 2019, Donoven and Harper showed that many Thompson groups are $\frac 32$-generated and posed five questions. The first of these is whether there exists a 2-generated group with every proper quotient cyclic that is not $\frac 32$-generated. This is a natural question given the significant work in proving that no finite group has this property, but we show that there is such an infinite group. The groups we consider are a family of finite index subgroups $G_1,\, G_2,\, \ldots$ of the Houghton group $\operatorname {FSym}(\mathbb {Z})\rtimes \mathbb {Z}$. We then show that $G_1$ and $G_2$ are $\frac 32$-generated and investigate the related notion of spread for these groups. We are able to show that they have finite spread at least 2. These are, therefore, the first infinite groups to be shown to have finite positive spread, and the first to be shown to have spread at least 2 (other than $\mathbb {Z}$ and the Tarski monsters, which have infinite spread). As a consequence, for each $k\in \{2,\, 3,\, \ldots \}$, we also have that $G_{2k}$ is index $k$ in $G_2$ but $G_2$ is $\frac 32$-generated whereas $G_{2k}$ is not.

Given an integer $g>2$ , we state necessary and sufficient conditions for a finite Abelian group to act as a group of automorphisms of some compact nonorientable Riemann surface of genus g. This result provides a new method to obtain the symmetric cross-cap number of Abelian groups. We also compute the least symmetric cross-cap number of Abelian groups of a given order and solve the maximum order problem for Abelian groups acting on nonorientable Riemann surfaces.