On the \(K\)-theory of boundary \(C^*\)-algebras of \(\tilde A_2\) groups
Let \(\mathbb K\) be a local field with residue field of order \(q\), and let \(\Gamma\) be an \(\widetilde{A_2}\) subgroup of \(\text{PGL}_3(\mathbb K)\). Let \(C(\mathbb P_{\mathbb K}^2,\mathbb Z)\) be the \(\Gamma\)-module of integer-valued continuous functions on the projective plane \(\mathbb P_{\mathbb K}^2\) over \(\mathbb K\).NEWLINENEWLINEIt is shown that the module of coinvariants \(C(\mathbb P_{\mathbb K}^2,\mathbb Z)_\Gamma=H_0(\Gamma;C(\mathbb P_{\mathbb K}^2,\mathbb Z))\) is finite and the class \([\mathbf 1]\) in \(C(\mathbb P_{\mathbb K}^2,\mathbb Z)_\Gamma \) has order bounded by \(q^2-1\). Its order equals \(q-1\) under the assumption that \(\Gamma\) is of Tits type and that \(q\neq 1\;(\text{mod}\,3)\).NEWLINENEWLINELet \(\Omega\) be the Furstenberg boundary of \(\text{PGL}_3(\mathbb K)\) and let \(C(\Omega)\rtimes\Gamma\) be the crossed product \(C^*\)-algebra. It is shown that the order of \([\mathbf 1]_{K_0}\) in the \(K_0\) group of \(C(\Omega)\rtimes\Gamma\) also equals \(q-1\) under the same assumption.
scientific article; zbMATH DE number 6072571