arXiv subjects

Bryan Jacobson

Publications and source records attributed to Bryan Jacobson.

2 recordsLinked to original sources

arXiv2019

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

math.GR↗

arXiv2017

Algebraic subgroups of acylindrically hyperbolic groups

A subgroup of a group $G$ is called 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.

math.GR↗