A mixed identity-free elementary amenable group

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 $\varphi: G \ast F_n \rightarrow G$suchthat$\varphi$istheidentityon$G$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.