The relative hyperbolicity of one-relator relative presentations

TL;DR

This paper proves that certain one-relator groups formed by adding a relator with exponent sum one to a free-torsion group are relatively hyperbolic with respect to the original group, expanding understanding of their geometric properties.

Contribution

It establishes the relative hyperbolicity of a class of one-relator groups constructed from free-torsion groups with specific relators.

Findings

Groups are relatively hyperbolic with respect to G

Relates to groups with relators of exponent sum one

Extends geometric group theory understanding

Abstract

We prove that if $G$ is a free-torsion group and $w(t)$ is a word in the alphabet $G \sqcup \{t^{\pm 1}\}$ with exponent sum one, then the group $<G,t|(w(t))^k = 1>$, where $k \geq 2$, is relatively hyperbolic with respect to $G$.