Boundary operator algebras for free uniform tree lattices

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Let X be a finite connected graph, each of whose vertices has degree at least three. The fundamental group Γ of X is a free group and acts on the universal covering tree Δ and on its boundary ∂ Δ, endowed with a natural topology and Borel measure. The crossed product C*-algebra C(∂ Δ) Γ depends only on the rank of Γ 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∞(∂ Δ) Γ is the hyperfinite factor of type IIIλ where λ=1/q² if X is bipartite, and λ=1/q otherwise.