Embeddings into highly transitive and mixed identity free groups
Group Theory
2025-09-15 v1
Abstract
Given a countable group , we develop a method to construct an overgroup that is finitely generated, highly transitive and mixed identity free. Our construction can be controlled to ensure that some fundamental group theoretic properties of are inherited by , such as amenability or the property of not containing a nonabelian free group. The former provides a strong solution to a question of Hull and Osin, and the latter provides the first examples of nonamenable groups without free subgroups that are highly transitive and mixed identity free. Our examples also have a nontrivial amenable radical, answering a question of Arzhantseva.
Cite
@article{arxiv.2509.09788,
title = {Embeddings into highly transitive and mixed identity free groups},
author = {James Hyde and Yash Lodha},
journal= {arXiv preprint arXiv:2509.09788},
year = {2025}
}