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.