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On lower bounds for cohomology growth in \(p\)-adic analytic towers

Mathematische Zeitschrift volume 277, pages 709–723 (2014)Cite this article

Abstract

Let \(p\) and \(\ell \) be two distinct prime numbers and let \(\Gamma \) be a group. We study the asymptotic behaviour of the mod-\(\ell \) Betti numbers in \(p\)-adic analytic towers of finite index subgroups. If \(\Theta \) is a finite \(\ell \)-group of automorphisms of \(\Gamma \), our main theorem allows to lift lower bounds for the mod-\(\ell \) cohomology growth in the fixed point group \(\Gamma ^\Theta \) to lower bounds for the growth in \(\Gamma \). We give applications to \(S\)-arithmetic groups and we also obtain a similar result for cohomology with rational coefficients.

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Notes

  1. Here “\(\gg \)“ means the inequality “\(\ge \)“ holds for large \(n\) up to a positive constant.

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  1. Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 , Bonn, Germany

    Steffen Kionke

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  1. Steffen Kionke
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Correspondence to Steffen Kionke.

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The author would like to thank the Max Planck Institute for Mathematics in Bonn for their hospitality and support.

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Kionke, S. On lower bounds for cohomology growth in \(p\)-adic analytic towers. Math. Z. 277, 709–723 (2014). https://doi.org/10.1007/s00209-013-1273-3

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