Abstract

The spectral gap of a symmetric stochastic matrix is the reciprocal of the best constant in its associated Poincare inequality. This inequality can be formulated in purely metric terms, where the metric is a Hilbertian metric. This immediately allows one to define the spectral gap of a matrix with respect to other, non-Euclidean, geometries: a standard procedure which is used a lot in embedding theory, most strikingly as a method to prove non-embeddability in the coarse category. Motivated by a combinatorial approach to the construction of bounded degree graph families which do not admit a coarse embedding into any uniformly convex normed space (such spaces were first constructed by Lafforgue), we will naturally arrive at questions related to the behavior of non-linear spectral gaps under graph operations such as powering and zig-zag products. We will also discuss the issue of constructing base graphs for these iterative constructions, which leads to new analytic and geometric challenges.