Approximating L^2 invariants of amenable covering spaces: A heat kernel approach

TL;DR

This paper introduces a heat kernel-based method to approximate L^2 invariants of amenable covering spaces, linking spectral and topological properties through regular exhaustions under certain conditions.

Contribution

It extends the principle of not feeling the boundary to heat kernels, enabling approximation of L^2 Betti numbers and spectral invariants in amenable coverings.

Findings

L^2 Betti numbers can be approximated by average Betti numbers of regular exhaustions.

Spectral invariants can be approximated by their averages over exhaustions.

The generalized principle of not feeling the boundary applies to heat kernels in this context.

Abstract

In this paper, we prove that the L^2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L^2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. The main tool which is used is a generalisation of the "principle of not feeling the boundary" (due to M. Kac), for heat kernels associated to boundary value problems.