On the growth of Betti numbers in \(p\)-adic analytic towers (Q399408)

Suppose that \(X\) is a connected compact CW-complex with fundamental group \(\Gamma\). Let \(p\) be a prime, and \(n\) a positive integer, and let \(\phi : \Gamma \rightarrow GL_{n}({\mathbb Z}_{p})\) be a homomorphism. The closure of the image of \(\phi\), denoted by \(G\), is a \(p\)-adic analytic group with an exhausting filtration by open normal subgroups NEWLINE\[NEWLINE G_{i} = ker(G \rightarrow GL_{n}({\mathbb Z}/p^{i}{\mathbb Z})) NEWLINE\]NEWLINE Let \(\Gamma_{i} = \phi^{-1}(G_{i})\), and let \(X_{i}\) be the corresponding finite cover of \(X\). Let \(\overline{X}\) be the cover of \(X\) corresponding to the kernel of \(\phi\) and let \(\overline{\Gamma} = \Gamma/Ker(\phi)\). This paper determines the growth of the Betti numbers NEWLINE\[NEWLINEb_{k}(X_{i}) = dim_{\mathbb Q}H_{k}(X_{i}, {\mathbb Q})NEWLINE\]NEWLINE NEWLINE\[NEWLINE b_{k}(X_{i}, {\mathbb F}_{p}) = dim_{{\mathbb F}_{p}}H_{k}(X_{i}, {{\mathbb F}_{p}})NEWLINE\]NEWLINENEWLINENEWLINESpecifically the authors show that, letting \(d= dim(G)\), for any integer \(k\) as \(i\) tends to infinity NEWLINE\[NEWLINEb_k(X_{i}) = \beta_{k}(\overline{X}, \overline{\Gamma})[\Gamma: \Gamma_{i}] + O([\Gamma : \Gamma_{i}]^{1-1/d}).NEWLINE\]NEWLINE Here \(\beta_{k}(\overline{X}, \overline{\Gamma})\) is the \(L^{2}\)-Betti number of the action of \(\overline{\Gamma}\) on \(\overline{X}\).