$p$-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 \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 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.