Malnormality and centers in one-relator relative presentations

TL;DR

This paper proves that in a specific group extension, the original group remains malnormal, leading to the conclusion that the extended group has a trivial center, which enhances understanding of group structure in one-relator relative presentations.

Contribution

It establishes the malnormality of the original group within a one-relator extension involving a proper power of a unimodular word.

Findings

G is malnormal in H

H has a trivial center if G is nontrivial

Provides structural insights into one-relator relative presentations

Abstract

Let H be a group obtained from a group G by adding one generator and one relator which is a proper power of a unimodular word. The main aim of this paper is to prove that G is a malnormal subgroup of H. This implies that if G is a nontrivial group, then the group H has a trivial center.