Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

TL;DR

This paper investigates the properties of groups formed by adding a generator and a relator that is a proper power, showing conditions under which these groups are relatively hyperbolic and contain free subgroups.

Contribution

It characterizes when one-relator relative presentations with powered unimodular relators are relatively hyperbolic and contain free subgroups, extending understanding of their algebraic structure.

Findings

Groups contain the free square of the initial group.

Almost always contain a non-abelian free subgroup.

Under certain conditions, groups are SQ-universal and relatively hyperbolic.

Abstract

A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent-sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.