Exactness of one relator groups (Q2781256)

Let \(\Gamma\) be a countable discrete group. \(\Gamma\) is called exact if for every short exact sequence \(0\rightarrow J\rightarrow A\rightarrow B\rightarrow 0\) of \(C^{*}\)-algebras the sequence \(0\rightarrow J\rtimes\Gamma\rightarrow A\rtimes\Gamma\rightarrow B\rtimes\Gamma\rightarrow 0\) is exact. \textit{E. Kirchberg} and \textit{S. Wassermann} [Math. Ann. 315, No. 2, 169--203 (1999; Zbl 0946.46054)] showed that \(\Gamma\) is \(C^{*}\)-exact if and only if its reduced \(C^{*}\)-algebra \(C_{r}^{*}(\Gamma)\) is exact.NEWLINENEWLINE\(\Gamma\) is said to be a one relator if it admits a presentation \(\langle X| R\rangle\) where \(X\) is a countable set and \(R\) is a single word over \(X\). Applying induction on the length of the relator \(R\), the author proves that every one relator group is \(C^{*}\)-exact. It is also shown that a countable discrete group \(\Gamma\) acting on a tree without inversion is \(C^{*}\)-exact if and only if the vertex stabilizers of the action are \(C^{*}\)-exact.NEWLINENEWLINEFor some similar results see \textit{J. L. Tu} [Bull. Soc. Math. Fr. 129, No. 1, 115--139 (2001; Zbl 1036.58021)].