Periodic Quotients of Free-like Groups

Let G be a free-like group, meaning that G is either non-elementary (word) hyperbolic or G is large (in the sense of Gromov). In this paper we suggest several approaches to construct continuous families of periodic quotients of G which enjoy 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 even 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 Kourovka Notebook.