$p$-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

and the limit in the field ${\mathbb{Q}}_p$ of $p$-adic numbers

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 $\chi ({\Gamma }_n,M)$ 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 $X=M^{\ast }$ 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.