Periodic quotients of hyperbolic and large groups

TL;DR

This paper develops multiple methods to construct continuous families of periodic quotients for hyperbolic and large groups, addressing longstanding questions and expanding understanding of their quotient structures.

Contribution

It introduces several new approaches for creating continuous families of periodic quotients of hyperbolic and large groups, including non-isomorphic and residually finite examples.

Findings

Constructed three continua of periodic quotients for hyperbolic groups.

Produced a continuum of non-isomorphic periodic residually finite quotients for large groups.

Provided positive answers to a question of Wiegold from Kourovka Notebook.

Abstract

Let $G$ be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this paper we describe several approaches for constructing continuous families of periodic quotients of $G$ with various properties. The first three methods work for any non-elementary hyperbolic group, producing three different continua of periodic quotients of $G$. They are based on the results and techniques, that were developed by Ivanov and Olshanskii in order to show that there exists an integer $n$ such that $G/G^n$ is an infinite group of exponent $n$. The fourth approach starts with a large group $G$ and produces a continuum of pairwise non-isomorphic periodic residually finite quotients. Speaking of a particular application, we use each of these methods to give a positive answer to a question of Wiegold from Kourovka Notebook.