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

TL;DR

This paper extends the Reidemeister-Schreier theorem to finitely $L$-presented groups, showing that finite index subgroups are also finitely $L$-presented and providing a constructive method for such presentations.

Contribution

It proves a variant of the Reidemeister-Schreier theorem for finitely $L$-presented groups and offers a constructive approach to obtain presentations of subgroups.

Findings

Finite index subgroups of finitely $L$-presented groups are finitely $L$-presented.

Provides a constructive method for obtaining $L$-presentations of subgroups.

Studies conditions for invariant $L$-presented subgroups to remain invariant.

Abstract

We prove a variant of the well-known Reidemeister-Schreier theorem for finitely $L$-presented groups. More precisely, we prove that each finite index subgroup of a finitely $L$-presented group is itself finitely $L$-presented. Our proof is constructive and it yields a finite $L$-presentation for the subgroup. We further study conditions on a finite index subgroup of an invariantly finitely $L$-presented group to be invariantly $L$-presented itself.