Homomorphisms to acylindrically hyperbolic groups I: Equationally noetherian groups and families

TL;DR

This paper investigates homomorphisms from finitely generated groups into uniformly acylindrically hyperbolic groups, establishing conditions under which these groups are equationally noetherian, with implications for relatively hyperbolic groups.

Contribution

It introduces a reduction technique for studying homomorphisms into acylindrically hyperbolic groups and proves that certain relatively hyperbolic groups are equationally noetherian.

Findings

Reduction of homomorphism sets to non-diverging cases

Relatively hyperbolic groups with noetherian peripherals are noetherian

Advances understanding of algebraic properties of acylindrically hyperbolic groups

Abstract

We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an appropriate sense. As an application, we prove that any relatively hyperbolic group with equationally noetherian peripheral subgroups is itself equationally noetherian.