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