Harmonic cochains and K-theory for ${\tilde{A}}_2$-groups

Abstract

If $\Gamma$ is a torsion free ${\tilde{A}}_2$-group acting on an ${\tilde{A}}_2$ building $\Delta$, and ${\mathfrak{A}}_{\Gamma }$ is the associated boundary $C^{\ast }$-algebra, it is proved that $K_0({\mathfrak{A}}_{\Gamma })\otimes \mathbb{R}\cong {\mathbb{R}}^{2{\beta }_2}$, where ${\beta }_2={\dim }_{\mathbb{R}}{H}^2(\Gamma ,\mathbb{R})$.