TORSION IN K-THEORY FOR BOUNDARY ACTIONS ON AFFINE BUILDINGS OF TYPE Ãn

Let Γ be a torsion free lattice in G = PGL(n+ 1,F), where n ≥ 1 and F is a non-archimedean local field. Then Γ acts on the Furstenberg boundary G/P , where P is a minimal parabolic subgroup of G. The identity element I in the crossed product C∗-algebra C(G/P ) ⋊ Γ generates a class [I] in the K0 group of C(G/P ) ⋊ Γ. It is shown that [I] is a torsion element of K0 and there is an explicit bound for the order of [I]. The result is proved more generally for groups acting on affine buildings of type Ãn. For n = 1, 2 the Euler-Poincaré characteristic χ(Γ) annihilates the class [I].