Definable equivalence relations and zeta functions of groups

2006/12/31 by Ehud Hrushovski, Ben Martin, Hrushovski, Ehud +5
Mathematics · #03C10 #03C60 #11M41 #20C15 #20E07 #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #math.GR #math.LO #msc:03C10 #msc:03C60 #msc:11M41 #msc:20C15 #msc:20E07

paper · pdf · doi:10.48550/arxiv.math/0701011

89 pages. Various corrections and changes. To appear in J. Eur. Math. Soc

arxiv created 2017/12/31 · arxiv updated 2018/01/03

Abstract

We prove that the theory of the p-adics \mathbb Q_p admits elimination of imaginaries provided we add a sort for \rm GL_n(\mathbb Q_p)/\rm GL_n(\mathbb Z_p) for each n. We also prove that the elimination of imaginaries is uniform in p. Using p-adic and motivic integration, we deduce the uniform rationality of certain formal zeta functions arising from definable equivalence relations. This also yields analogous results for definable equivalence relations over local fields of positive characteristic. The appendix contains an alternative proof, using cell decomposition, of the rationality (for fixed p) of these formal zeta functions that extends to the subanalytic context. As an application, we prove rationality and uniformity results for zeta functions obtained by counting twist isomorphism classes of irreducible representations of finitely generated nilpotent groups; these are analogous to similar results of Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroup zeta functions of finitely generated nilpotent groups.

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