Algebraic subgroups of acylindrically hyperbolic groups
Bryan Jacobson

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 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 of an acylindrically hyperbolic group is algebraic if and only if there exists a finite subgroup of such that . 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.
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