Boundary operator algebras for free uniform tree lattices

Let $X$ be a finite connected graph, each of whose vertices has degree at least three. The fundamental group $\Gamma$ of $X$ is a free group and acts on the universal covering tree $\Delta$ and on its boundary $\partial \Delta$, endowed with a natural topology and Borel measure. The crossed product $C^*$-algebra $C(\partial \Delta) \rtimes \Gamma$ depends only on the rank of $\Gamma$ and is a Cuntz-Krieger algebra whose structure is explicitly determined. The crossed product von Neumann algebra does not possess this rigidity. If $X$ is homogeneous of degree $q+1$ then the von Neumann algebra $L^\infty(\partial \Delta)\rtimes \Gamma$ is the hyperfinite factor of type $III_\lambda$ where $\lambda=1/{q^2}$ if $X$ is bipartite, and $\lambda=1/{q}$ otherwise.