Exactness of one relator groups

Abstract

A discrete groupΓ{\Gamma }isC∗C^*-exact if the reduced crossed product withΓ{\Gamma }converts a short exact sequence ofΓ{\Gamma }-C∗C^*-algebras into a short exact sequence ofC∗C^*-algebras. A one relator group is a discrete groupΓ{\Gamma }admitting a presentationΓ=⟨X|R⟩{\Gamma }=\langle \; X \;|\; R \;\ranglewhereXXis a countable set andRRis a single word overXX. In this short paper we prove that all one relator discrete groups areC∗C^*-exact. Using the Bass-Serre theory we also prove that a countable discrete groupΓ\Gammaacting without inversion on a tree isC∗C^*-exact if the vertex stabilizers of the action areC∗C^*-exact.