p-adic limits of renormalized logarithmic Euler characteristics

Given a countable residually finite group $\mathrm{\Gamma }$, we write ${\mathrm{\Gamma }}_n\to e$ if $({\mathrm{\Gamma }}_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of ${\mathrm{\Gamma }}_n$’s contains only the unit element $e$ of $\mathrm{\Gamma }$. Given a $\mathrm{\Gamma }$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n , M)|^{(-1)^i} \end{equation} and the limit in the field ${\mathbb{Q}}_p$ of $p$-adic numbers \begin{equation} h_p := \lim_{n\to\infty} (\Gamma : \Gamma_n)^{-1} \log_p \chi (\Gamma_n , M) ; . \end{equation} Here ${\log }_p:{\mathbb{Q}}_p^{\times }\to {\mathbb{Z}}_p$ 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 $\chi ({\mathrm{\Gamma }}_n,M)$ are well defined. That notion is a $p$-adic analogue of expansiveness of the dynamical system given by the $\mathrm{\Gamma }$-action on the compact Pontrjagin dual $X=M^{\ast }$ of $M$. Under further conditions on $\mathrm{\Gamma }$ 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 $L^2$ $R$-torsion of a $\mathrm{\Gamma }$-module $M$ which they related to the entropy $h$ of the $\mathrm{\Gamma }$-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 $\mathrm{\Gamma }={\mathbb{Z}}^N$ in more detail where our theory is related to Serre’s intersection numbers on arithmetic schemes.