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Eigenvalues of random walks on groups. (English) Zbl 0852.60078

Summary: 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 a new geometric method. 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.

MSC:

60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
60G50 Sums of independent random variables; random walks
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