Two projects on equations over generalizations of hyperbolic groups

Abstract

In this dissertation, we present the results of two projects, both related to equations over groups and each concerning a particular generalization of hyperbolic groups. Their abstracts are presented below. A subgroup of a group $G$ is called \textit{algebraic} if it can be expressed as a finite union of solution sets to systems of equations. We prove that a non-elementary subgroup $H$ of an acylindrically hyperbolic group $G$ is algebraic if and only if there exists a finite subgroup $K$ of $G$ such that $C_G(K) \leq H \leq N_G(K)$. We provide some applications of this result to free products, torsion-free relatively hyperbolic groups, and ascending chains of algebraic subgroups in acylindrically hyperbolic groups. A group $G$ is called \textit{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 provided by Ol'shanskii, Osin, and Sapir in their paper \textit{Lacunary hyperbolic groups} 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.