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arXiv · 1410.1693

On computing homology gradients over finite fields

Łukasz Grabowski·Thomas Schick
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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.

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Łukasz Grabowski, Thomas Schick. 2016-06-06. On computing homology gradients over finite fields. https://doi.org/10.1017/s0305004116000657

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