Remarks on approximability and stability for groups

TL;DR

This paper explores how approximation and stability properties of groups are preserved under certain quotient operations, especially involving finitely generated normal subgroups with Kazhdan's property (T), with applications to geometric group theory.

Contribution

It demonstrates new inheritance results for approximation and stability properties in groups, particularly through quotients with specific normal subgroups, extending previous understanding.

Findings

Inheritance of approximation properties in quotients with finitely generated normal subgroups

Stability properties preserved when quotienting by normal subgroups with Kazhdan's property (T)

Applications to geometric group constructions using Wise and Belegradek--Osin variations

Abstract

In this paper, we provide several instances in which interesting approximation and stability properties are inherited by quotients with respect to finitely generated normal subgroups or, more strongly, normal subgroups with Kazhdan's property (T). Applications arise when these observations are combined with variations of the Rips construction due to Wise and Belegradek--Osin.