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, five or seven squares have residually finite fundamental group. The smallest open cases were CSCs with four squares and directed complete V H complexes with six squares. We prove that the CSC with four squares studied by Janzen and Wise has a non-residually finite fundamental group. 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. 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 .

Small non-Leighton two-complexes

“…of the free abelian group and the Baumslag-Solitar group BS (3,5) contains no subgroups isomorphic to the Klein-bottle group K = BS(1, −1).…”

“…The group BS (3,5) does not contain subgroups isomorphic to K [11] and is torsionfree. Therefore, applying once again (1), we obtain that the quotient G/ [a, G] = a ∞ × BS (3,5) by the normal closure [a, G] of the set [a, G] of commutators of a and all elements of G has no nonidentity elements conjugate their inverse. Therefore, any element of G conjugate to its inverse lies in N = [a, G] .…”

“…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 i as finite-index subgroups, but, in 1 , any finite-index subgroup contains a finite-index subgroup which is the direct product of free groups, while 2 has no such finite-index subgroups [8] ( 2 is not even residually finite [3], [5]). The results of [8] imply 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.…”

Bireversible automata generating lamplighter groups

“…[12]). The interplay between groups of bireversible automata and fundamental groups of corresponding square complexes was studied in [5].…”

“…We also present here required background on square complexes and their connection with automata. Our presentation is based on [8] and [5] where one can find omitted details. In Section 3 for arbitrary finite abelian group we define corresponding square complex and permutational automaton and present their basic properties.…”