Random walks on groups

This thesis is concerned with random walks on solvable matrix groups, direct products of automorphism groups of trees, semi-direct products arising from totally disconnected locally compact groups and unrestricted lamplighter groups. Brofferio and Schapira, described the Poisson boundary of GL_n(ℚ) for measures of finite first moment with respect to adelic length. We define matrix groups FG_n(P) for each natural number n and finite set of primes P, such that every rational-valued upper triangular matrix group is a (possibly distorted) subgroup. We show that adelic length is a word metric estimate on FG_n(P)by constructing another, intermediate, word metric estimate which can be easily computed from the entries of any matrix in the group. Finite first moment of a probability measure with respect to adelic length is equivalent to finite first moment with respect to word length in FG_n(P). The Poisson boundaries of finite direct products of affine automorphism groups of homogeneous trees are also considered. The Poisson boundary is a product of ends of trees with a hitting measure for spread-out, aperiodic measures of finite first moment, whose closed support generates subgroups which are not fully exceptional. The Poisson boundary of a semi-direct product, V−− ⋊ α, for any automorphism α and tidy compact open subgroup V in a locally compact, totally disconnected group G is also shown to be the space of ends of the tree with the hitting measure under similar assumptions. Boundary triviality is discussed in both cases. This extends work of Cartwright, Kaimanovich and Woess. In the final chapter, we discuss pointwise convergence and non-trivial boundaries for unrestricted lamplighter groups. We define a rate of eschewal on the rough Cayley graph of a compactly generated, totally disconnected, locally compact group G. For appropriate choices of compact open subgroups, the rate of eschewal is finite and equal to the rate of escape for measures supported within the restricted lamplighter subgroup.