Automaton groups and complete square complexes

Abstract

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for short) with twelve squares. At the same time, Janzen and Wise proved that CSCs with at most three squares, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete VH complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. In particular, this gives a nonresidually finite CAT(0) group isometric to F 2 ×F 2 . For the class of complete directed VH complexes, we prove that there are exactly two complexes with six squares having a non-residually finite fundamental group. In particular, this positively answers to a question of Wise on whether the main example from his PhD thesis is non-residually finite. As a by-product, we get finitely presented torsion-free simple groups which decompose into an amalgamated free product of free groups F 7 * F 49 F 7 .Our approach relies on the connection between square complexes and automata discovered by Glasner and Mozes, where complete VH complexes with one vertex correspond to bireversible automata. We prove that the square complex associated to a bireversible automaton with two states or over the binary alphabet generating an infinite automaton group has a non-residually finite fundamental group. We describe automaton groups associated to CSCs with four squares and get two simple automaton representations of the free group F 2 and the first automaton representation of the free product C 3 * C 3 .

Almost-automorphisms of trees, cloning systems and finiteness properties

“…Since finite groups are of type F ∞ , we have the following example, which is worth remarking on first. Other examples of self-similar groups of type F ∞ that have been considered in the literature include free groups [VV10,SVV11], free products of arbitrarily many copies of Z/2Z [SV11], and (Z/3Z) * (Z/3Z) [BK17]. In all these cases, the groups are not only self-similar but are even so called automaton groups.…”

Finite and Infinite Quotients of Discrete and Indiscrete Groups

“…Although the presentations of Γ SV and Γ JW are rather similar, the groups are quite different. Indeed Γ SV is linear in characteristic 3, while the group Γ JW fails to be residually finite (this was observed independently in [BK,Theorem 15] and [CW]; see also Section 4.6 below).…”

“…In fact, D. Wise's results allow one to derive more precise information in the case of BMWgroups. The following assertion will be relevant to our purposes (see [BK,Theorem 9] for a related statement). We denote by A * 2 the set of all words of the form ab with a, b ∈ A ∪ A −1 .…”

Small non-Leighton two-complexes

“…The both of these complex are covered by the Cartesian product of two trees (Cayley graphs of the free group F 2 ); and no finite common cover exists, because the fundamental group of such hypothetical covering complex would embed in both groups Γ i as finite-index subgroups, but, in Γ 1 , any finite-index subgroup contains a finite-index subgroup which is the direct product of free groups, while Γ 2 has no such finite-index subgroups [JaW09] (Γ 2 is not even residually finite [CaW18], [BoK21]). The results of [JaW09] implies also a minimality of this example in the sense that if we restrict ourselves to complexes K i covered by products of trees, then four two-dimensional cells is the minimum among all non-Leighton pairs.…”