Eigenvalues of the Natural Random Walk on the Burnside Group $B(3, n)$

Abstract

In this paper we give sharp bounds on the eigenvalues of the natural random walk on the Burnside group $B(3,n)$. Most of the argument uses established geometric techniques for eigenvalue bounds. However, the most interesting bound, the upper bound on the second largest eigenvalue, cannot be done by existing techniques. To give a bound we use a novel method for bounding the eigenvalues of a random walk on a group $G$ (or equivalently its Cayley graph). This method works by choosing eigenvectors which fall into representations of an Abelian normal subgroup of $G$. One is then left with a large number (one for each representation) of easier problems to analyze.