On finite state automaton actions of HNN extensions of free abelian groups

Abstract

HNN extensions of free abelian groups are considered. For arbitrary prime $p$ it is introduced a class of such extensions that act by finite automaton permutations over an alphabet $ \mathsf{X} $ of cardinality $p$ and belong to $p$-Sylow subgroup of the group of automaton permutations over such $ \mathsf{X} $. As a corollary it implies that all corresponding HNN extensions are residually $p$-finite.

On exponentiation, p-automata and HNN extensions of free abelian groups

“…Since all groups deőned by őnite p-automata are residually p-őnite (see e.g. [13]) the second statement now immediately follows.…”

“…In the present paper we consider this problem for HNN extensions of free abelian groups to extend results of [13]. We use automata constructed in [1] such that their groups are required HNN extensions.…”

The Self-Similarity of Free Semigroups and Groups

“…In addition, there are also results for amalgamated products and HNN extensions of nonfinite groups. For example, Lavrenyuk, Mazorchuk, Oliynyk and Sushchansky presented free products of two infinite cyclic groups with amalgamation over an infinite cyclic subgroup as a subgroup of an automaton group [22] and Prokhorchuk considered HNN extensions of certain free abelian groups [30]. With regard to self-similar presentations, Vorobets and Vorobets generalized the Bellaterra automaton to present any free product of an odd number of groups of order two as an automaton group [35, Theorem 1.7 (i)]; as with the Aleshin automaton, these automata can be combined to also cover (sufficiently large) even numbers of copies of the group of order two [35,Theorem 1.7 (ii)].…”

On a finite state representation of GL(n,Z)

“…An algorithm for constructing a finite state automorphism by a given unimodular matrix is discussed and implemented. Finally, a method from [1,9] is applied to construct a quite surprising representation of the free group of rank 2 by finite state automorphisms of a regular rooted tree based on the representation of GL(2, Z) (cf. with Section 7 of [17]).…”