Presentations of Groups with Even Length Relations

TL;DR

This paper investigates groups with presentations where generators square to identity and all relations are even length, exploring their structure, parabolic subgroups, and the non-uniqueness of element factorizations.

Contribution

It characterizes the structure of such groups, analyzes parabolic subgroups, and provides a counterexample showing non-uniqueness of factorizations.

Findings

Every element admits a factorization relative to a parabolic subgroup.

Counterexample demonstrates that such factorizations are not always unique.

Provides insights into the properties of groups with even length relations.

Abstract

We study the properties of groups that have presentations in which the square of each generator gives the identity and all relations are of even length. We consider the parabolic subgroups of such a group and show that every element has a factorisation with respect to a given parabolic subgroup. Furthermore, we give a counterexample, using a cluster group presentation, which demonstrates that this factorisation is not necessarily unique.