On the ET0L subgroup membership problem in bounded automata groups

TL;DR

This paper proves that in bounded automata groups, the subgroup membership problem for stabilisers of eventually periodic infinite rays can be characterized as an ET0L language, providing a constructive solution and insights into the group's structure.

Contribution

It establishes that the membership problem for certain stabilisers in bounded automata groups is an ET0L language, extending understanding of subgroup structures in automata groups.

Findings

Membership problem is ET0L for eventually periodic rays

The problem cannot be generally context-free in this setting

Provides a recursive formula for the Green function on Schreier graphs

Abstract

We are interested in the subgroup membership problem in groups acting on rooted $d$-regular trees and a natural class of subgroups, the stabilisers of infinite rays emanating from the root. These rays, which can also be viewed as infinite words in the alphabet with d letters, form the boundary of the tree. Stabilisers of infinite rays are not finitely generated in general, but if the ray is computable, the membership problem is well posed and solvable. The main result of the paper is that, for bounded automata groups, the membership problem in the stabiliser of any ray that is eventually periodic as an infinite word, forms an ET0L language that is constructable. The result is optimal in the sense that, in general, the membership problem for the stabiliser of an infinite ray in a bounded automata group cannot be context-free. As an application, we give a recursive formula for the associated generating function, aka the Green function, on the corresponding infinite Schreier graph.