Rights statement: https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society The final, definitive version of this article has been published in the Journal, Mathematical Proceedings of the Cambridge Philosophical Society, 162 (3), pp 507-532 2017, © 2016 Cambridge University Press.
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Research output: Contribution to Journal/Magazine › Journal article › peer-review
Research output: Contribution to Journal/Magazine › Journal article › peer-review
}
TY - JOUR
T1 - On computing homology gradients over finite fields
AU - Grabowski, Łukasz
AU - Schick, Thomas
N1 - https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society The final, definitive version of this article has been published in the Journal, Mathematical Proceedings of the Cambridge Philosophical Society, 162 (3), pp 507-532 2017, © 2016 Cambridge University Press.
PY - 2017/5
Y1 - 2017/5
N2 - 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.
AB - 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.
KW - math.GT
KW - math.GR
U2 - 10.1017/S0305004116000657
DO - 10.1017/S0305004116000657
M3 - Journal article
VL - 162
SP - 507
EP - 532
JO - Mathematical Proceedings of the Cambridge Philosophical Society
JF - Mathematical Proceedings of the Cambridge Philosophical Society
SN - 0305-0041
IS - 3
ER -