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 $C^*$-algebra, it is proved that $K_0(\mathfrak{A}_\Gamma)\otimes \mathbb{R} \cong \mathbb{R}^{2\beta_2}$, where $\beta_2=\dim_{\mathbb{R}} H^2(\Gamma, \mathbb{R})$.