$p$-Adic limits of renormalized logarithmic Euler characteristics

Abstract

Given a countable residually finite group $\Gamma$, we write $\Gamma\_n \to e$ if $(\Gamma\_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $\Gamma\_n$'s contains only the unit element $e$ of $\Gamma$. Given a $\Gamma$-module $M$ we are interested in the multiplicative Euler characteristics $$ \label{eq:1a} \chi (\Gamma\_n , M) = \prod\_i |H\_i (\Gamma\_n , M)|^{(-1)^i} $$ and the limit in the field $\mathbb Q\_p$ of $p$-adic numbers $$ \label{eq:1b} h\_p := \lim\_{n\to\infty} (\Gamma : \Gamma\_n)^{-1} \log\_p \chi (\Gamma\_n , M) ; . $$ Here $\log\_p : \mathbb Q^{\times}\_p \to \mathbb Z\_p$ is the branch of the $p$-adic logarithm with $\log\_p (p) = 0$. Of course, neither expression will exist in general. We isolate conditions on $M$, in particular $p$-adic expansiveness which guarantee that the Euler characteristics $\chi (\Gamma\_n,M)$ are well defined. That notion is a $p$-adic analogue of expansiveness of the dynamical system given by the $\Gamma$-action on the compact Pontrjagin dual $X=M^∗$ of $M$. Under further conditions on $\Gamma$ we also show that the renormalized $p$-adic limit in the second formula exists and equals the $p$-adic $R$-torsion of $M$. The latter is a $p$-adic analogue of the Li–Thom L2 $R$-torsion of a $\Gamma$-module $M$ which they related to the entropy h of the $\Gamma$-action on $X$. We view the limit $h\_p$ as a version of entropy which values in the $p$-adic numbers and the equality with $p$-adic $R$-torsion as an analogue of the Li–Thom formula in the expansive case. We discuss the case $\Gamma = \mathbb{Z}^n$ in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.