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