Related papers: Automaton groups and complete square complexes

We consider Turing machines as actions over configurations in $\Sigma^{\mathbb{Z}^d}$ which only change them locally around a marked position that can move and carry a particular state. In this setting we study the monoid of Turing machines…

We show that any topological, closed, oriented, non-spin $4$-manifold with fundamental group $\mathbb{Z}_{4k}$ and $\min(b_2^+, b_2^-)\geq 15$, has either none or infinitely many distinct smooth structures. Furthermore, we construct…

It is shown that there exist finitely generated infinite simple groups of infinite commutator width and infinite square width on which there exists no stably unbounded conjugation-invariant norm, and in particular stable commutator length…

We construct recursively-presented finitely-generated torsion groups which have bounded torsion and whose word problem is conjunctive equivalent (in particular positive and Turing equivalent) to a given recursively enumerable set. These…

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.

In this note we prove that, for a finite semigroup $S$, the dual Cayley automaton semigroup $\mathbf{C^{\ast}}(S)$ is finite if and only if $S$ is $\mathcal{H}$-trivial and has no non-trivial right zero subsemigroups.

We prove that a random group has fixed points when it isometrically acts on a CAT(0) cube complex. We do not assume that the action is simplicial.

Let $H$ be a finite quasisimple classical group, i.e. $H$ is perfect and $S:=H/Z(H)$ is a finite simple classical group. We prove in this paper that, excluding the cases when the simple group $S$ has a very exceptional Schur multiplier such…

A finite group $G$ is a called a DCI-group if any two isomorphic Cayley digraphs of $G$ are also isomorphic via an automorphism of $G$. If $G$ is a non-abelian generalised dihedral DCI-group, then Dobson, Muzychuk, and Spiga proved that $G$…

The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is…

In this article we study the automorphism group ${\rm Aut}(X,\sigma)$ of subshifts $(X,\sigma)$ of low word complexity. In particular, we prove that Aut$(X,\sigma)$ is virtually $\mathbb{Z}$ for aperiodic minimal subshifts and certain…

In this paper we explore the structure and properties of C-groups. We define a C-group as a group $G$ with $rk(G) < rk(Z(G))$ (where $rk(G)$ is the minimal cardinal of a generating set for a group $G$). Using GAP (a group theory program)…

We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in…

In this work, we study three families of locally graded groups with finitely many orbits under automorphisms. We prove that: (i) a residually finite group with finitely many orbits under automorphisms is locally finite and has finite…

We show that groups presented by inverse-closed finite convergent length-reducing rewriting systems are characterised by a striking geometric property: their Cayley graphs are geodetic and side-lengths of non-degenerate triangles are…

We study a class of two-generator two-relator groups, denoted $J_n(m,k)$, that arise in the study of relative asphericity as groups satisfying a transitional curvature condition. Particular instances of these groups occur in the literature…

The paper was motivated by a question of Vilonen, and the main results have been used by Mirkovic and Vilonen to give a geometric interpretation of the dual group (as a Chevalley group over Z) of a reductive group. We define a…

We show that plane Cremona groups over finite fields embed as dense subgroups into Neretin groups, i.e. groups of almost automorphisms of rooted trees. We also show that if the finite base field has even characteristic and contains at least…

We provide lower estimates on the minimal number of generators of the profinite completion of free products of finite groups. In particular, we show that if C_1,...,C_n are finite cyclic groups then there exists a finite group G which is…

For any given finite group, Schulte and Williams (2015) establish the existence of a convex polytope whose combinatorial automorphisms form a group isomorphic to the given group. We provide here a shorter proof for a stronger result: the…