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

TL;DR

This paper investigates how Betti numbers grow in p-adic analytic towers of covers, offering simplified proofs of known approximation results and extending some findings to pro-p towers.

Contribution

It provides simplified proofs of Betti number growth in p-adic towers and extends results to pro-p towers, broadening the understanding of their asymptotic behavior.

Findings

Simplified proofs of Betti number approximation results

Extension of results to arbitrary pro-p towers

Partial results on Betti number growth in pro-p 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.