Approximating the Schur multiplier of certain infinitely presented groups via nilpotent quotients

Abstract

We describe an algorithm for computing successive quotients of the Schur multiplier M (G) of a group G given by an invariant finite L-presentation. As applications, we investigate the Schur multipliers of various self-similar groups, including the Grigorchuk super-group, the generalized Fabrykowski-Gupta groups, the Basilica group and the Brunner-Sidki-Vieira group.

A Reidemeister-Schreier theorem for finitely $L$-presented groups

“…Finite L-presentations allow computer algorithms to be applied in the investigation of the groups they define. For instance, they allow to compute the lower central series quotients [2], the Dwyer quotients of the group's Schur multiplier [15], and even a coset-enumeration process exists for finitely L-presented groups [13]. It is the aim of this paper to prove the following variant of Reidemeister-Schreier's theorem: Theorem 1.1 Each finite index subgroup of a finitely L-presented group is finitely L-presented.…”

A Note on Invariantly Finitely $L$-Presented Groups

“…Finite L-presentations allow computer algorithms to be applied in the investigation of the groups they define. For instance, they allow one to compute the lower central series quotients [2], to compute the Dwyer quotients of the group's Schur multiplier [10], to develop a coset enumerator for finite index subgroups [11], and even the Reidemeister-Schreier theorem for finitely presented groups generalizes to finitely L-presented groups [13]. For a survey on the application of computers in the investigation of finitely L-presented groups, we refer to [12].…”

Investigating self-similar groups using their finite $L$-presentation

“…A first algorithm for finitely L-presented groups is the nilpotent quotient algorithm [31,5]. Recently, further algorithms for finitely L-presented groups were developed [33,34,35]. For instance, in [34], a coset enumeration process for finitely L-presented groups was described.…”