JournalsggdVol. 14, No. 2pp. 427–467

pp-Adic limits of renormalized logarithmic Euler characteristics

  • Christopher Deninger

    Universität Münster, Germany
$p$-Adic limits of renormalized logarithmic Euler characteristics cover
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Abstract

Given a countable residually finite group Γ\Gamma, we write Γne\Gamma_n \to e if (Γn)(\Gamma_n) is a sequence of normal subgroups of finite index such that any infinite intersection of Γn\Gamma_n's contains only the unit element ee of Γ\Gamma. Given a Γ\Gamma-module MM we are interested in the multiplicative Euler characteristics

\labeleq:1aχ(Γn,M)=iHi(Γn,M)(1)i\label{eq:1a} \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n , M)|^{(-1)^i}

and the limit in the field Qp\mathbb Q_p of pp-adic numbers

\labeleq:1bhp:=limn(Γ:Γn)1logpχ(Γn,M)  .\label{eq:1b} h_p := \lim_{n\to\infty} (\Gamma : \Gamma_n)^{-1} \log_p \chi (\Gamma_n , M) \; .

Here logp:Qp×Zp\log_p : \mathbb Q^{\times}_p \to \mathbb Z_p is the branch of the pp-adic logarithm with logp(p)=0\log_p (p) = 0. Of course, neither expression will exist in general. We isolate conditions on MM, in particular pp-adic expansiveness which guarantee that the Euler characteristics χ(Γn,M)\chi (\Gamma_n,M) are well defined. That notion is a pp-adic analogue of expansiveness of the dynamical system given by the Γ\Gamma-action on the compact Pontrjagin dual X=MX=M^∗ of MM. Under further conditions on Γ\Gamma we also show that the renormalized pp-adic limit in the second formula exists and equals the pp-adic RR-torsion of MM. The latter is a pp-adic analogue of the Li–Thom L2 RR-torsion of a Γ\Gamma-module MM which they related to the entropy h of the Γ\Gamma-action on XX. We view the limit hph_p as a version of entropy which values in the pp-adic numbers and the equality with pp-adic RR-torsion as an analogue of the Li–Thom formula in the expansive case. We discuss the case Γ=Zn\Gamma = \mathbb{Z}^n in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.

Cite this article

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

DOI 10.4171/GGD/550