# Periodic quotients of hyperbolic and large groups

### Ashot Minasyan

University of Southampton, United Kingdom### Alexander Yu. Olshanskii

Vanderbilt University, Nashville, United States### Dmitriy Sonkin

University of Virginia, Charlottesville, United States

## Abstract

Let G be either a non-elementary (word) hyperbolic group or a large group (both in the sense of Gromov). In this article 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/Gn 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 the Kourovka Notebook.

## Cite this article

Ashot Minasyan, Alexander Yu. Olshanskii, Dmitriy Sonkin, Periodic quotients of hyperbolic and large groups. Groups Geom. Dyn. 3 (2009), no. 3, pp. 423–452

DOI 10.4171/GGD/65