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Given a countable residually finite group , we write if is a sequence of normal subgroups of finite index such that any infinite intersection of 's contains only the unit element of . Given a -module we are interested in the multiplicative Euler characteristics
and the limit in the field of -adic numbers
Here is the branch of the -adic logarithm with . Of course, neither expression will exist in general. We isolate conditions on , in particular -adic expansiveness which guarantee that the Euler characteristics are well defined. That notion is a -adic analogue of expansiveness of the dynamical system given by the -action on the compact Pontrjagin dual of . Under further conditions on we also show that the renormalized -adic limit in the second formula exists and equals the -adic -torsion of . The latter is a -adic analogue of the Li–Thom L2 -torsion of a -module which they related to the entropy h of the -action on . We view the limit as a version of entropy which values in the -adic numbers and the equality with -adic -torsion as an analogue of the Li–Thom formula in the expansive case. We discuss the case in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.
Cite this article
Christopher Deninger, -Adic limits of renormalized logarithmic Euler characteristics. Groups Geom. Dyn. 14 (2020), no. 2, pp. 427–467DOI 10.4171/GGD/550