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

TL;DR

This paper demonstrates that the $L^2$ Betti numbers of amenable covering spaces can be approximated using combinatorial methods, confirming a previous conjecture and establishing determinant class properties.

Contribution

It introduces a combinatorial approach to approximate $L^2$ invariants of amenable coverings, proving a conjecture and showing these spaces are of determinant class.

Findings

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

Amenable covering spaces are of determinant class

Confirmed a previously made conjecture

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, proving a conjecture that we made in an earlier paper. We also prove that an arbitrary amenable covering space of a finite simplicial complex is of determinant class.