On computing homology gradients over finite fields

TL;DR

This paper explores a method for computing homology gradients over finite fields, revealing their potential irrationality and variability across different fields, extending previous work on l^2-Betti numbers.

Contribution

It demonstrates that the spectral measure method can be used to compute homology gradients over arbitrary fields, and shows these can be irrational or vary infinitely across fields.

Findings

Homology gradients over fields of characteristic not 2 can be irrational.

Existence of a CW-complex with infinitely many distinct homology gradients over different fields.

The spectral measure method applies to a broad class of homology gradient computations.

Abstract

Recently the so-called Atiyah conjecture about l^2-Betti numbers has been disproved. The counterexamples were found using a specific method of computing the spectral measure of a matrix over a complex group ring. We show that in many situations the same method allows to compute homology gradients, i.e. generalizations of l^2-Betti numbers to fields of arbitrary characteristic. As an application we point out that (i) the homology gradient over any field of characteristic different than 2 can be an irrational number, and (ii) there exists a finite CW-complex with the property that the homology gradients of its universal cover taken over different fields have infinitely many different values.