On the growth of Betti numbers in $p$-adic analytic towers

Abstract

We study the asymptotic growth of Betti numbers in tower of finite covers and provide simple proofs of approximation results, which were previously obtained by Calegari-Emerton, in the generality of arbitrary p-adic analytic towers of covers. Further, we also obtain partial results about arbitrary pro-$p$ towers.

15 pages, the previous version contained an in the proof of Theorem 2.1 that is now corrected. This is the last version, it will appear in GGD