Definable equivalence relations and zeta functions of groups

Ehud Hrushovski, Ben Martin, Silvain Rideau, Raf Cluckers

Research output: Contribution to journalArticle

4 Citations (Scopus)
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Abstract

We prove that the theory of the p-adics Qp admits elimination of imaginaries
provided we add a sort for GLn(Qp)/GLn(Zp) 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.
Original languageEnglish
Pages (from-to)2467-2537
Number of pages71
JournalJournal of the European Mathematical Society
Volume20
Issue number10
DOIs
Publication statusPublished - 18 Jul 2018

Fingerprint

Equivalence relation
Rationality
Riemann zeta function
Finitely Generated Group
Nilpotent Group
Motivic Integration
Cell Decomposition
Isomorphism Classes
Positive Characteristic
Local Field
P-adic
Twist
Uniformity
Sort
Elimination
Deduce
Counting
Decomposition
Alternatives

Keywords

  • Elimination of imaginaries
  • invariant extensions of types
  • cell decompositions
  • rational zeta functions
  • subgroup zeta functions
  • representation zeta functions

Cite this

Definable equivalence relations and zeta functions of groups. / Hrushovski, Ehud; Martin, Ben; Rideau, Silvain; Cluckers, Raf.

In: Journal of the European Mathematical Society, Vol. 20, No. 10, 18.07.2018, p. 2467-2537.

Research output: Contribution to journalArticle

Hrushovski, Ehud ; Martin, Ben ; Rideau, Silvain ; Cluckers, Raf. / Definable equivalence relations and zeta functions of groups. In: Journal of the European Mathematical Society. 2018 ; Vol. 20, No. 10. pp. 2467-2537.
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keywords = "Elimination of imaginaries, invariant extensions of types, cell decompositions, rational zeta functions, subgroup zeta functions, representation zeta functions",
author = "Ehud Hrushovski and Ben Martin and Silvain Rideau and Raf Cluckers",
note = "The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.",
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N1 - The authors wish to thank Thomas Rohwer, Deirdre Haskell, Dugald Macpherson and Elisabeth Bouscaren for their comments on earlier drafts of this work, Martin Hils for suggesting that the proof could be adapted to finite extensions and Zo´e Chatzidakis for pointing out an error in how constants were handled in earlier versions. The second author is grateful to Jamshid Derakhshan, Marcus du Sautoy, Andrei Jaikin-Zapirain, Angus Macintyre, Dugald Macpherson, Mark Ryten, Christopher Voll and Michele Zordan for helpful conversations. We are grateful to Alex Lubotzky for suggesting studying representation growth; several of the ideas in Section 8 are due to him. The first author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC Grant agreement no. 291111/ MODAG, the second author was supported by a Golda Meir Postdoctoral Fellowship at the Hebrew University of Jerusalem and the third author was partly supported by ANR MODIG (ANR-09-BLAN-0047) Model Theory and Interactions with Geometry. The author of the appendix would like to thank M. du Sautoy, C. Voll, and Kien Huu Nguyen for interesting discussions on this and related subjects. He was partially supported by the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013) with ERC Grant Agreement nr. 615722 MOTMELSUM and he thanks the Labex CEMPI (ANR-11-LABX-0007-01). We are grateful to the referee for their careful reading of the paper and for their many comments, corrections and suggestions for improving the exposition. In memory of Fritz Grunewald.

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N2 - We prove that the theory of the p-adics Qp admits elimination of imaginariesprovided we add a sort for GLn(Qp)/GLn(Zp) for each n. We also prove that theelimination of imaginaries is uniform in p. Using p-adic and motivic integration,we deduce the uniform rationality of certain formal zeta functions arising fromdefinable equivalence relations. This also yields analogous results for definableequivalence relations over local fields of positive characteristic. The appendixcontains an alternative proof, using cell decomposition, of the rationality (forfixed p) of these formal zeta functions that extends to the subanalytic context.As an application, we prove rationality and uniformity results for zeta functionsobtained by counting twist isomorphism classes of irreducible representationsof finitely generated nilpotent groups; these are analogous to similar resultsof Grunewald, Segal and Smith and of du Sautoy and Grunewald for subgroupzeta functions of finitely generated nilpotent groups.

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