ABSTRACT
Let G be the vertex set of a homogeneous, infinite tree of finite degree. Let
{RXy)x
ye
c be the transition matrix of a random walk on G which proceeds
along edges of the tree. Suppose that R is a symmetric matrix, so that there is
a probability assigned to each undirected edge. Suppose also that the walk is
vertex homogeneous, i.e. that the probabilities assigned to the edges leaving a
given vertex form a multiset which is independent of the vertex.
The group G Z2 * Z2 * Z2 acts simply transitively on G, leaving
the random walk invariant. Thus R commutes with the action of G on
£2(G).
For a e sp(i2) we construct H
a
, the generalized cr-eigenspace of R, which is
a representation space for G. We analytically continue these principal series
representations to find complementary series representations. Our main result
is that these representations are (with a few exceptions) irreducible.
Key words and phrases, homogeneous tree, free group, algebraicity, nearest-neighbor, ran-
dom walk, unitary representation, principal series, complementary series, boundary realization,
irreducibility, inequivalence.
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