Coset enumeration for certain infinitely presented groups

TL;DR

This paper introduces an algorithm to compute the index of finite index subgroups in finitely L-presented groups, demonstrating decidability of subgroup membership and analyzing low-index subgroups in specific self-similar groups.

Contribution

It presents a novel algorithm for index computation and subgroup membership decision in finitely L-presented groups, with applications to self-similar groups.

Findings

Algorithm successfully computes subgroup indices when finite

Decidability of subgroup membership problem established

Analysis of low-index subgroups in specific self-similar groups

Abstract

We describe an algorithm that computes the index of a finitely generated subgroup in a finitely $L$-presented group provided that this index is finite. This algorithm shows that the subgroup membership problem for finite index subgroups in a finitely $L$-presented group is decidable. As an application, we consider the low-index subgroups of some self-similar groups including the Grigorchuk group, the twisted twin of the Grigorchuk group, the Grigorchuk super-group and the Hanoi 3-group