A mixed identity-free elementary amenable group

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 $\phi :G\ast F_n\to G$ such that $\phi$ is the identity on $G$ and $\phi (w)$ 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.