Algebraic subgroups of acylindrically hyperbolic groups

TL;DR

This paper characterizes algebraic subgroups of acylindrically hyperbolic groups, showing they are precisely those containing the centralizer of a finite subgroup and contained in its normalizer, with applications to various group classes.

Contribution

It provides a complete characterization of algebraic subgroups in acylindrically hyperbolic groups, linking them to finite subgroups and their centralizers and normalizers.

Findings

Algebraic subgroups are characterized by finite subgroups K with C_G(K) ≤ H ≤ N_G(K).

Applications include free products and torsion-free relatively hyperbolic groups.

Results describe ascending chains of algebraic subgroups in these groups.

Abstract

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.