We show that there is a distortion element in a finitely generated subgroup G of the automorphism group of the full shift, namely an element of infinite order whose word norm grows polylogarithmically. As a corollary, we obtain a lower bound on the entropy dimension of any subshift containing a copy of G, and that a sofic shift’s automorphism group contains a distortion element if and only if the sofic shift is uncountable. We obtain also that groups of Turing machines and the higher-dimensional Brin–Thompson groups $mV$ admit distortion elements; in particular, $2V$ (unlike V) does not admit a proper action on a CAT$(0)$ cube complex. In each case, the distortion element roughly corresponds to the SMART machine of Cassaigne, Ollinger, and Torres-Avilés [A small minimal aperiodic reversible Turing machine. J. Comput. System Sci. 84 (2017), 288–301].

Several finite complex reflection groups have a braid group that is isomorphic to a torus knot group. The reflection group is obtained from the torus knot group by declaring meridians to have order k for some $k\geq 2$, and meridians are mapped to reflections. We study all possible quotients of torus knot groups obtained by requiring meridians to have finite order. Using the theory of J-groups of Achar and Aubert [‘On rank 2 complex reflection groups’, Comm. Algebra 36(6) (2008), 2092–2132], we show that these groups behave like (in general, infinite) complex reflection groups of rank two. The large family of ‘toric reflection groups’ that we obtain includes, among others, all finite complex reflection groups of rank two with a single conjugacy class of reflecting hyperplanes, as well as Coxeter’s truncations of the $3$-strand braid group. We classify these toric reflection groups and explain why the corresponding torus knot group can be naturally considered as its braid group. In particular, this yields a new infinite family of reflection-like groups admitting braid groups that are Garside groups. Moreover, we show that a toric reflection group has cyclic center by showing that the quotient by the center is isomorphic to the alternating subgroup of a Coxeter group of rank three. To this end we use the fact that the center of the alternating subgroup of an irreducible, infinite Coxeter group of rank at least three is trivial. Several ingredients of the proofs are purely Coxeter-theoretic, and might be of independent interest.

We generalise some known results for limit groups over free groups and residually free groups to limit groups over Droms RAAGs and residually Droms RAAGs, respectively. We show that limit groups over Droms RAAGs are free-by-(torsion-free nilpotent). We prove that if S is a full subdirect product of type $FP_s(\mathbb{Q})$ of limit groups over Droms RAAGs with trivial center, then the projection of S to the direct product of any s of the limit groups over Droms RAAGs has finite index. Moreover, we compute the growth of homology groups and the volume gradients for limit groups over Droms RAAGs in any dimension and for finitely presented residually Droms RAAGs of type $FP_m$ in dimensions up to m. In particular, this gives the values of the analytic $L^2$-Betti numbers of these groups in the respective dimensions.

In 1968, Steinberg [Endomorphisms of Linear Algebraic Groups, Memoirs of the American Mathematical Society, 80 (American Mathematical Society, Providence, RI, 1968)] proved a theorem stating that the exterior powers of an irreducible reflection representation of a Euclidean reflection group are again irreducible and pairwise nonisomorphic. We extend this result to a more general context where the inner product invariant under the group action may not necessarily exist.

In this paper, we study triple-product-free sets, which are analogous to the widely studied concept of product-free sets. A nonempty subset S of a group G is triple-product-free if $abc \notin S$ for all $a, b, c \in S$. If S is triple-product-free and is not a proper subset of any other triple-product-free set, we say that S is locally maximal. We classify all groups containing a locally maximal triple-product-free set of size 1. We then derive necessary and sufficient conditions for a subset of a group to be locally maximal triple-product-free, and conclude with some observations and comparisons with the situation for standard product-free sets.

Let $W_{\Gamma} $ be the right-angled Coxeter group with defining graph $\Gamma $. We show that the asymptotic dimension of $W_{\Gamma} $ is smaller than or equal to $\mathrm{dim}_{CC}(\Gamma )$, the clique-connected dimension of the graph. We generalize this result to graph products of finite groups.

We construct finitely generated torsion-free solvable groups G that have infinite rank, but such that all finitely generated torsion-free metabelian subquotients of G are virtually abelian. In particular all finitely generated metabelian subgroups of G are virtually abelian. The existence of such groups shows that there is no “torsion-free version” of P. Kropholler’s theorem, which characterises solvable groups of infinite rank via their metabelian subquotients.

A hyperbolic group G acts by homeomorphisms on its Gromov boundary. We show that if $\partial G$ is a topological n–sphere, the action is topologically stable in the dynamical sense: any nearby action is semi-conjugate to the standard boundary action.

We prove a Banach version of Żuk’s criterion for groups acting on partite (i.e., colorable) simplicial complexes. Using this new criterion, we derive a new fixed point theorem for random groups in the Gromov density model with respect to several classes of Banach spaces ($L^p$ spaces, Hilbertian spaces, uniformly curved spaces). In particular, we show that for every p, a group in the Gromov density model has asymptotically almost surely property $(F L^p)$ and give a sharp lower bound for the growth of the conformal dimension of the boundary of such group as a function of the parameters of the density model.

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.

In the setting of finite groups, suppose $J$ acts on $N$ via automorphisms so that the induced semidirect product $N\rtimes J$ acts on some non-empty set $\Omega$, with $N$ acting transitively. Glauberman proved that if the orders of $J$ and $N$ are coprime, then $J$ fixes a point in $\Omega$. We consider the non-coprime case and show that if $N$ is abelian and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$. We also show that if $N$ is nilpotent, $N\rtimes J$ is supersoluble, and a Sylow $p$-subgroup of $J$ fixes a point in $\Omega$ for each prime $p$, then $J$ fixes a point in $\Omega$.

We study the Eisenstein series associated to the full rank cusps in a complete hyperbolic manifold. We show that given a Kleinian group $\Gamma <{\operatorname{\mathrm{Isom}}}^+(\mathbb H^{n+1})$, each full rank cusp corresponds to a cohomology class in $H^{n}(\Gamma , V)$, where V is either the trivial coefficient or the adjoint representation. Moreover, by computing the intertwining operator, we show that different cusps give rise to linearly independent classes.

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.

A well-known theorem of Philip Hall states that if a group G has a nilpotent normal subgroup N such that $G/N'$ is nilpotent, then G itself is nilpotent. We say that a group class 𝔛 is a Hall class if it contains every group G admitting a nilpotent normal subgroup N such that $G/N'$ belongs to 𝔛. Hall classes have been considered by several authors, such as Plotkin [‘Some properties of automorphisms of nilpotent groups’, Soviet Math. Dokl. 2 (1961), 471–474] and Robinson [‘A property of the lower central series of a group’, Math. Z. 107 (1968), 225–231]. A further detailed study of Hall classes is performed by us in another paper [‘Hall classes of groups’, to appear] and we also investigate the behaviour of the class of finite-by-𝔜 groups for a given Hall class 𝔜 [‘Hall classes in linear groups’, to appear]. The aim of this paper is to prove that for most natural choices of the Hall class 𝔜, also the classes $(\mathbf{L}\mathfrak{F})\mathfrak{Y}$ and 𝔅𝔜 are Hall classes, where L𝔉 is the class of locally finite groups and 𝔅 is the class of locally finite groups of finite exponent.

A left orderable monster is a finitely generated left orderable group all of whose fixed point-free actions on the line are proximal: the action is semiconjugate to a minimal action so that for every bounded interval I and open interval J, there is a group element that sends I into J. In his 2018 ICM address, Navas asked about the existence of left orderable monsters. By now there are several examples, all of which are finitely generated but not finitely presentable. We provide the first examples of left orderable monsters that are finitely presentable, and even of type $F_\infty $. These groups satisfy several additional properties separating them from the previous examples: they are not simple, they act minimally on the circle, and they have an infinite-dimensional space of homogeneous quasimorphisms. Our construction is flexible enough that it produces infinitely many isomorphism classes of finitely presented (and type $F_{\infty }$) left orderable monsters.

Let G be a group that is either virtually soluble or virtually free, and let ω be a weight on G. We prove that if G is infinite, then there is some maximal left ideal of finite codimension in the Beurling algebra $\ell^1(G, \omega)$, which fails to be (algebraically) finitely generated. This implies that a conjecture of Dales and Żelazko holds for these Banach algebras. We then go on to give examples of weighted groups for which this property fails in a strong way. For instance, we describe a Beurling algebra on an infinite group in which every closed left ideal of finite codimension is finitely generated and which has many such ideals in the sense of being residually finite dimensional. These examples seem to be hard cases for proving Dales and Żelazko’s conjecture.

We construct finitely generated groups of small period growth, i.e. groups where the maximum order of an element of word length n grows very slowly in n. This answers a question of Bradford related to the lawlessness growth of groups and is connected to an approximative version of the restricted Burnside problem.

We study a family of finitely generated residually finite small-cancellation groups. These groups are quotients of $F_2$ depending on a subset $S$ of positive integers. Varying $S$ yields continuously many groups up to quasi-isometry.