On the K-theory of boundary $C^{\ast }$-algebras of ${\tilde{A}}_2$ groups

Abstract

Let $\Gamma$ be an ${\tilde{A}}_2$ subgroup of $\PGL_3(\mathbb K)$, where $\mathbb{K}$ is a local field with residue field of order $q$. The module of coinvariants $C({\mathbb{P}}_{\mathbb{K}}^2,\mathbb{Z}{)}_{\Gamma }$ is shown to be finite, where ${\mathbb{P}}_{\mathbb{K}}^2$ is the projective plane over $\mathbb{K}$. If the group $\Gamma$ is of Tits type and if $q\not\equiv 1(\mathrm{mod}3)$ then the exact value of the order of the class $[I{]}_{K_0}$ in the K-theory of the (full) crossed product $C^{\ast }$-algebra $C(\Omega )⋊\Gamma$ 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.

References in corpus (1)

• Euler characteristics and Gysin sequences for group actions on boundaries