Pattern closure of groups of tree automorphisms

TL;DR

This paper characterizes the topological closure of certain self-similar groups of tree automorphisms by pattern restrictions and demonstrates the non-existence of specific finitely constrained groups in binary trees.

Contribution

It establishes a connection between pattern-avoiding groups and self-similar, regular branch groups, providing new insights into their structure and constraints.

Findings

Closure of pattern-avoiding groups is a self-similar, regular branch group.

No infinite, finitely constrained, topologically finitely generated groups exist with forbidden patterns of size two in binary trees.

Abstract

It is shown that a group defined by forbidding all patterns of size s+1 that do not appear in a given self-similar group of tree automorphisms is the topological closure of a self-similar, countable, regular branch group, branching over its level s stabilizer. As an application, it is shown that there are no infinite, finitely constrained, topologically finitely generated groups of binary tree automorphisms defined by forbidden patterns of size two.