Let Γ be an Ã2 subgroup of PGL3(), where is a local field with residue field of order q. The module of coinvariants C(,ℤ)Γ is shown to be finite, where is the projective plane over . If the group Γ is of Tits type and if q ≢ 1 (mod 3) then the exact value of the order of the class [1]K0 in the K-theory of the (full) crossed product C*-algebra C(Ω) ⋊ Γ is determined, where Ω is the Furstenberg boundary of PGL3(). For groups of Tits type, this verifies a conjecture of G. Robertson and T. Steger.

This work is concerned with type theory for reductive groups over a non Archimedean field. Given such a field F, and a division algebra D of finite dimension over its center F, we obtain results concerning the construction of simple types for the group GL(m, D), $m\geqslant1$. More precisely, for each simple stratum of the matrix algebra M(m, D), we produce a set of β-extensions in the sense of Bushnell and Kutzko.

Let $\Gamma$ be a torsion free lattice in $G=\text{PGL}\left( 3,\mathbb{F} \right)$ where $\mathbb{F}$ is a nonarchimedean local field. Then $\Gamma$ acts freely on the affine Bruhat-Tits building $B$ of $G$ and there is an induced action on the boundary $\Omega$ of $B$. The crossed product ${{C}^{*}}$ -algebra $\mathcal{A}\left( \Gamma \right)=C\left( \Omega \right)\rtimes \Gamma$ depends only on $\Gamma$ and is classified by its $K$-theory. This article shows how to compute the $K$-theory of $\mathcal{A}\left( \Gamma \right)$ and of the larger class of rank two Cuntz-Krieger algebras.