Harmonic Analysis for Anisotropic Random Walks on Homogeneous Trees

This work presents a detailed study of the anisotropic series representations of the free product group $\mathbf Z/2\mathbf Z\star \cdots \star \mathbf Z/2\mathbf Z$. These representations are infinite dimensional, irreducible, and unitary and can be divided into principal and complementary series. Anisotropic series representations are interesting because, while they are not restricted from any larger continuous group in which the discrete group is a lattice, they nonetheless share many properties of such restrictions. The results of this work are also valid for nonabelian free groups on finitely many generators.