Eigenvalues of random walks on groups

In this paper we discuss and apply a novel method for bounding the eigenvalues of a random walk on a group G (or equivalently on its Cayley graph). This method works by looking at the action of an Abelian normal subgroup H of G on G. We may then choose eigenvectors which fall into representations of H. One is then left with a large number (one for each representation of H) of easier problems to analyze. This analysis is carried out by new geometric methods. This method allows us to give bounds on the second largest eigenvalue of random walks on nilpotent groups with low class number. The method also lets us treat certain very easy solvable groups and to give better bounds for certain nice nilpotent groups with large class number. For example, we will give sharp bounds for two natural random walks on groups of upper triangular matrices.