Related papers: More on Groups and Counter Automata

We describe the autotopism group Atp(G) of any abelian group G as being a semidirect product of its automorphism group Aut(G) and G^2. We then provide the subgroup structure of Atp(G) when G is a finite cyclic group.

Generalizing the idea of self-similar groups defined by Mealy automata, we itroduce the notion of a self-similar automaton and a self-similar group over a changing alphabet. We show that every finitely generated residually-finite group is…

We will show that every element of a finitely generated abelian group is automorphically equivalent what we will define to be a {\em representative element} in a {\em repeat-free subgroup}, and for finite abelian groups we can count the…

A result of Baumslag and Roseblade states that a finitely presented subgroup of the direct product of two free groups is virtually a direct product of free groups. In this paper we generalise this result to the class of cyclic subgroup…

We show that any Kahler extension of a finitely generated abelian group by a surface group of genus g at least 2 is virtually a product. Conversely, we prove that any homomorphism of an even rank, finitely generated abelian group into the…

If G and H are finitely generated, residually nilpotent metabelian groups, H is termed para-G if there is a homomorphism of G into H which induces an isomorphism between the corresponding terms of their lower central quotient groups. We…

In this paper we describe finitely generated groups $H$ universally equivalent (with constants from $G$ in the language) to a given torsion-free relatively hyperbolic group $G$ with free abelian parabolics. It turns out that, as in the free…

A finitely generated group is said to be an automata group if it admits a faithful self-similar finite-state representation on some regular $m$-tree. We prove that if $G$ is a subgroup of an automata group, then for each finitely generated…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

We prove that every finitely generated soluble group which is not virtually abelian has a subgroup of one of a small number of types.

This work is a continuation of Automorphisms of $K$-groups I, P. Flavell, preprint. The main object of study is a finite $K$-group $G$ that admits an elementary abelian group $A$ acting coprimely. For certain group theoretic properties…

The notion of an automaton over a changing alphabet $X=(X_i)_{i\geq 1}$ is used to define and study automorphism groups of the tree $X^*$ of finite words over $X$. The concept of bi-reversibility for Mealy-type automata is extended to…

We prove that the semigroup generated by a finite state Mealy automaton $\mathcal{A}=(Q,A,\tau)$ is infinite if and only if there exists some right-infinite word in the alphabet $A$ with infinite orbit.

In this paper, we study the word problem for automaton semigroups and automaton groups from a complexity point of view. As an intermediate concept between automaton semigroups and automaton groups, we introduce automaton-inverse semigroups,…

(Free-abelian)-by-free, self-similar groups generated by finite self-similar sets of tree automorphisms and having unsolvable conjugacy problem are constructed. Along the way, orbit undecidable, free subgroups of GL_d(Z), for d > 5, and…

We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index…

Let $d \geq 2$ be an integer. We conjecture that there is a finitely generated perfect group whose homomorphic images include all finite $d$-generated perfect groups. We prove a special case of this conjecture for the finite perfect groups…

Given a finite abelian group $G$ and elements $x, y \in G$, we prove that there exists $\phi \in \text{Aut}(G)$ such that $\phi(x) = y$ if and only if $G/\langle x \rangle \cong G/\langle y \rangle$. This result leads to our development of…

We explore a natural class of semigroups that have word problem decidable by finite state automata. Among the main results are invariance of this property under change of generators, invariance under basic algebraic constructions and…

In this article we construct asynchronous and sometimes synchronous automatic structures for amalgamated products and HNN extensions of groups that are strongly asynchronously (or synchronously) coset automatic with respect to the…