On computing homology gradients over finite fields
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.
Keywords
Cite
@article{arxiv.1410.1693,
title = {On computing homology gradients over finite fields},
author = {Łukasz Grabowski and Thomas Schick},
journal= {arXiv preprint arXiv:1410.1693},
year = {2016}
}
Comments
30 pages, 6 figures, v3: changes in exposition, fixed proof of Theorem 4.1. To appear in Cambridge Math. Proc