A mixed identity-free elementary amenable group

TL;DR

This paper constructs finitely generated elementary amenable groups that are mixed identity-free, extending previous work on lacunary hyperbolic groups, and also produces locally finite p-groups with this property.

Contribution

It introduces a modified construction method for elementary amenable groups to be mixed identity-free, expanding the class of known such groups.

Findings

Constructed finitely generated elementary amenable mixed identity-free groups.

Produced locally finite p-groups that are mixed identity-free.

Extended existing constructions of lacunary hyperbolic groups.

Abstract

A group $G$ is called mixed identity-free if for every $n \in \mathbb{N}$ and every $w \in G \ast F_n$ there exists a homomorphism $\varphi: G \ast F_n \rightarrow G$ such that $\varphi$ is the identity on $G$ and $\varphi(w)$ is nontrivial. In this paper, we make a modification to the construction of elementary amenable lacunary hyperbolic groups given by Ol'shanskii, Osin, and Sapir to produce finitely generated elementary amenable groups which are mixed identity-free. As a byproduct of this construction, we also obtain locally finite $p$-groups which are mixed identity-free.