Exactness of one relator groups

Abstract:

A discrete group ${\Gamma }$ is $C^*$-exact if the reduced crossed product with ${\Gamma }$ converts a short exact sequence of ${\Gamma }$-$C^*$-algebras into a short exact sequence of $C^*$-algebras. A one relator group is a discrete group ${\Gamma }$ admitting a presentation ${\Gamma }=\langle \; X \;|\; R \;\rangle$ where $X$ is a countable set and $R$ is a single word over $X$. In this short paper we prove that all one relator discrete groups are $C^*$-exact. Using the Bass-Serre theory we also prove that a countable discrete group $\Gamma$ acting without inversion on a tree is $C^*$-exact if the vertex stabilizers of the action are $C^*$-exact.