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 CG(K)≤H≤NG(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.