- 1410.1693v3
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- https://www.cambridge.org/core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society/article/on-computing-homology-gradients-over-finite-fields/0F82153951F14AD5055444EDCBE7D184
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**On computing homology gradients over finite fields.** / Grabowski, Łukasz; Schick, Thomas.

Research output: Contribution to journal › Journal article › peer-review

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

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

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

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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.",

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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.",

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AU - Schick, Thomas

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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.

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