p-adic limits of renormalized logarithmic Euler characteristics

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Given a countable residually finite group Γ, we write Γₙ → e if (Γₙ) is a sequence of normal subgroups of finite index such that any infinite intersection of Γₙ's contains only the unit element e of Γ. Given a Γ-module M we are interested in the multiplicative Euler characteristics equation χ(Γₙ, M) = ∏ᵢ |Hᵢ (Γₙ, M)|⁽⁻¹⁾ⁱ equation and the limit in the field Qₚ of p-adic numbers equation hₚ:= ₙ→∞ (Γ: Γₙ)⁻¹ ₚ χ(Γₙ, M). equation Here ₚ: Q×ₚ → Zₚ is the branch of the p-adic logarithm with 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(Γₙ, 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 hₚ 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= Zⁿ in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.