Harmonic cochains and K-theory for $\widetilde A_2$ groups

If $Γ$ is a torsion free $\widetilde A_2$ group acting on an $\widetilde A_2$ building $Δ$, and $\fk A_Γ$ is the associated boundary $C^*$-algebra, it is proved that $K_0(\fk A_Γ)\otimes \bb R \cong \bb R^{2β_2}$, where $β_2=\dim_\bb R H^2(Γ, \bb R)$.

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