On the permanence properties of residually exact groups

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