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• 1410.1693v3

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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## On computing homology gradients over finite fields

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### Standard

On computing homology gradients over finite fields. / Grabowski, Łukasz; Schick, Thomas.

In: Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 162, No. 3, 05.2017, p. 507-532.

Research output: Contribution to journalJournal articlepeer-review

### Harvard

Grabowski, Ł & Schick, T 2017, 'On computing homology gradients over finite fields', Mathematical Proceedings of the Cambridge Philosophical Society, vol. 162, no. 3, pp. 507-532. https://doi.org/10.1017/S0305004116000657

### APA

Grabowski, Ł., & Schick, T. (2017). On computing homology gradients over finite fields. Mathematical Proceedings of the Cambridge Philosophical Society, 162(3), 507-532. https://doi.org/10.1017/S0305004116000657

### Vancouver

Grabowski Ł, Schick T. On computing homology gradients over finite fields. Mathematical Proceedings of the Cambridge Philosophical Society. 2017 May;162(3):507-532. https://doi.org/10.1017/S0305004116000657

### Author

Grabowski, Łukasz ; Schick, Thomas. / On computing homology gradients over finite fields. In: Mathematical Proceedings of the Cambridge Philosophical Society. 2017 ; Vol. 162, No. 3. pp. 507-532.

### Bibtex

@article{a976890d69614854af7e9232d00a83b5,
title = "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 = "math.GT, math.GR",
author = "{\L}ukasz Grabowski and Thomas Schick",
note = "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, {\textcopyright} 2016 Cambridge University Press.",
year = "2017",
month = may,
doi = "10.1017/S0305004116000657",
language = "English",
volume = "162",
pages = "507--532",
journal = "Mathematical Proceedings of the Cambridge Philosophical Society",
issn = "0305-0041",
publisher = "Cambridge University Press",
number = "3",

}

### RIS

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 -