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On computing homology gradients over finite fields

Published online by Cambridge University Press:  05 August 2016

ŁUKASZ GRABOWSKI  and
THOMAS SCHICK
ŁUKASZ GRABOWSKI
Affiliation:
Department of Mathematics and Statistics, Lancaster University, Lancaster LA1 4YF.
THOMAS SCHICK
Affiliation:
Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3, 37073 Göttingen, Germany.
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Abstract

Recently the so-called Atiyah conjecture about l2-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. generalisations of l2-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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 3 , May 2017 , pp. 507 - 532
Copyright
Copyright © Cambridge Philosophical Society 2016 

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