## The Annals of Probability

### Eigenvalues of Random Walks on Groups

Richard Stong

#### Abstract

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.

#### Article information

Source
Ann. Probab., Volume 23, Number 4 (1995), 1961-1981.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176987811

Digital Object Identifier
doi:10.1214/aop/1176987811

Mathematical Reviews number (MathSciNet)
MR1379176

Zentralblatt MATH identifier
0852.60078

JSTOR