On the permanence properties of residually exact groups

TL;DR

This paper investigates the class of residually exact groups, showing it remains stable under certain group constructions, thus advancing understanding of their algebraic and analytical properties.

Contribution

It proves that residually exact groups are closed under Green's graph products, double amalgamated products, and special HNN extensions.

Findings

Residually exact groups are closed under Green's graph products.

Residually exact groups are closed under double amalgamated products.

Residually exact groups are closed under special HNN extensions.

Abstract

A discrete group $\Gamma$ is called exact if the reduced group C*-algebra ${C_{\lambda}}^{*}(\Gamma)$ is exact as C*-algebras, and a discrete group $\Lambda$ is called residually exact if every nonunital element $g \in \Lambda$ admits a surjective group homomorphism from $\Lambda$ to some exact group $\Gamma$ which maps $g$ to a nonunital element of $\Gamma$. We prove the class of residually exact groups is closed under taking Green's graph products [1], double amalgamed products and special HNN extensions.