A Note on Invariantly Finitely $L$-Presented Groups

TL;DR

This paper extends Tietze transformations to invariantly finitely $L$-presented groups, proving their property independence from generating sets and generalizing normal subgroup results in finitely presented groups.

Contribution

It introduces Tietze transformations for $L$-presentations and generalizes invariance and subgroup results beyond previous limitations.

Findings

Tietze transformations for $L$-presentations are established.

Invariance of being finitely $L$-presented is independent of generating set.

Normal subgroups with certain quotient conditions are invariantly finitely $L$-presented.

Abstract

In the first part of this note, we introduce Tietze transformations for $L$-presentations. These transformations enable us to generalize Tietze's theorem for finitely presented groups to invariantly finitely $L$-presented groups. Moreover, they allow us to prove that `being invariantly finitely $L$-presented' is an abstract property of a group which does not depend on the generating set. In the second part of this note, we consider finitely generated normal subgroups of finitely presented groups. Benli proved that a finitely generated normal subgroup of a finitely presented group is invariantly finitely $L$-presented whenever its quotient is infinite cyclic. We generalize this result to the case where the finitely presented group splits over its finitely generated subgroup and to the case where the quotient is abelian with torsion-free rank at most two.