On the K-theory of boundary $C^*$-algebras of $\widetilde A_2$ groups

Abstract

Let $\Gamma$ be an $\widetilde A_2$ subgroup of $\PGL_3(\mathbb K)$, where $\mathbb K$ is a local field with residue field of order $q$. The module of coinvariants $C(\mathbb P^2_{\mathbb K},\mathbb Z)_{\Gamma}$ is shown to be finite, where $\mathbb P^2_{\mathbb K}$ is the projective plane over $\mathbb K$. If the group $\Gamma$ is of Tits type and if $q \not\equiv 1 \pmod {3}$ then the exact value of the order of the class $[I]_{K_0}$ in the K-theory of the (full) crossed product $C^*$-algebra $C(\Omega)\rtimes\Gamma$ is determined, where $\Omega$ is the Furstenberg boundary of $\PGL_3(\mathbb K)$. For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.