If
Γ
is a torsion free
A
˜
2
group acting on an
A
˜
2
building
Δ
, and $\fk A_{\Gamma}$ is the associated boundary
C
∗
-algebra, it is proved that $K_0(\fk A_\Gamma)\otimes \bb R \cong \bb R^{2\beta_2}$, where $\beta_2=\dim_\bb R H^2(\Gamma, \bb R)$.