Coarse non-amenability and covers with small eigenvalues

TL;DR

This paper constructs a tower of finite covers of a Riemannian manifold with fundamental group surjecting onto a free group, where the first non-zero Laplacian eigenvalues tend to zero, demonstrating non-amenable coarse behavior.

Contribution

It provides the first example of such a tower not obtainable by previous methods requiring an amenable fundamental group surjection.

Findings

First non-zero eigenvalues (M_n) tend to zero as n increases.

Constructs towers with non-amenable coarse properties.

Shows limitations of previous methods based on amenability.

Abstract

Given a closed Riemannian manifold M and a (virtual) epimorphism from the fundamental group of M onto a free group of rank 2, we construct a tower of finite sheeted regular covers {M_n}_{n=0}^{\infty} of M such that the first non-zero eigenvalues \lambda_1(M_n) of the Laplacian converge to zero as n tends to infinity. This is the first example of such a tower which is not obtainable up to uniform quasi-isometry (or even up to uniform coarse equivalence) by the previously known methods where the fundamental group of M is supposed to surject onto an amenable group.