# $p$-Adic limits of renormalized logarithmic Euler characteristics

### Christopher Deninger

Universität Münster, Germany

## Abstract

Given a countable residually finite group $Γ$, we write $Γ_{n}→e$ if $(Γ_{n})$ is a sequence of normal subgroups of finite index such that any infinite intersection of $Γ_{n}$'s contains only the unit element $e$ of $Γ$. Given a $Γ$-module $M$ we are interested in the multiplicative Euler characteristics

and the limit in the field $Q_{p}$ of $p$-adic numbers

Here $g_{p}:Q_{p}→Z_{p}$ is the branch of the $p$-adic logarithm with $g_{p}(p)=0$. Of course, neither expression will exist in general. We isolate conditions on $M$, in particular $p$-adic expansiveness which guarantee that the Euler characteristics $χ(Γ_{n},M)$ are well defined. That notion is a $p$-adic analogue of expansiveness of the dynamical system given by the $Γ$-action on the compact Pontrjagin dual $X=M_{∗}$ of $M$. Under further conditions on $Γ$ 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 $Γ$-module $M$ which they related to the entropy h of the $Γ$-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 $Γ=Z_{n}$ in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.

## Cite this article

Christopher Deninger, $p$-Adic limits of renormalized logarithmic Euler characteristics. Groups Geom. Dyn. 14 (2020), no. 2, pp. 427–467

DOI 10.4171/GGD/550