Grabowski, Łukasz and Schick, Thomas
(2017)
*On computing homology gradients over finite fields.*
Mathematical Proceedings of the Cambridge Philosophical Society, 162 (3).
pp. 507-532.
ISSN 0305-0041

1410.1693v3.pdf - Accepted Version

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## 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.