THE MARTIN BOUNDARY FOR HARMONIC-FUNCTIONS ON GROUPS OF AUTOMORPHISMS OF A HOMOGENEOUS TREE

Consider a closed subgroup Gamma of the automorphism group of a homogeneous tree T, and assume that Gamma acts transitively on the vertex set. Suppose that mu is a probability measure on Gamma which has continuous density with respect to Haar measure and whose support is compact open and generates Gamma as a closed semigroup. It is shown that the Martin boundary of Gamma with respect to the random walk with law mu coincides with the space of ends of T. This extends known results for free groups and applies, for example, to the affine group over a non Archimedean local field.