On representation zeta functions of groups and a conjecture of Larsen-Lubotzky

Nir Avni; Benjamin Klopsch; Uri Onn; Christopher Voll

doi:10.1016/j.crma.2010.02.019

On representation zeta functions of groups and a conjecture of Larsen-Lubotzky

Nir Avni^*, Benjamin Klopsch, Uri Onn, Christopher Voll

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A₂ defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

Original language	English (US)
Pages (from-to)	363-367
Number of pages	5
Journal	Comptes Rendus Mathematique
Volume	348
Issue number	7-8
DOIs	https://doi.org/10.1016/j.crma.2010.02.019
State	Published - Apr 2010

Funding

The authors, in various constellations, would like to thank Alex Lubotzky, Odile Sauzet and the following institutions for their support: the Batsheva de Rothschild Fund for the Advancement of Science, the EPSRC, the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation. E-mail addresses: [email protected] (N. Avni), [email protected] (B. Klopsch), [email protected] (U. Onn), [email protected] (C. Voll). 1 Supported by NSF grant DMS-0901638.

ASJC Scopus subject areas

General Mathematics

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10.1016/j.crma.2010.02.019

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title = "On representation zeta functions of groups and a conjecture of Larsen-Lubotzky",

abstract = "We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.",

author = "Nir Avni and Benjamin Klopsch and Uri Onn and Christopher Voll",

note = "Funding Information: The authors, in various constellations, would like to thank Alex Lubotzky, Odile Sauzet and the following institutions for their support: the Batsheva de Rothschild Fund for the Advancement of Science, the EPSRC, the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation. Funding Information: E-mail addresses: [email protected] (N. Avni), [email protected] (B. Klopsch), [email protected] (U. Onn), [email protected] (C. Voll). 1 Supported by NSF grant DMS-0901638.",

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doi = "10.1016/j.crma.2010.02.019",

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AU - Avni, Nir

AU - Klopsch, Benjamin

AU - Onn, Uri

AU - Voll, Christopher

N1 - Funding Information: The authors, in various constellations, would like to thank Alex Lubotzky, Odile Sauzet and the following institutions for their support: the Batsheva de Rothschild Fund for the Advancement of Science, the EPSRC, the Mathematisches Forschungsinstitut Oberwolfach and the Nuffield Foundation. Funding Information: E-mail addresses: [email protected] (N. Avni), [email protected] (B. Klopsch), [email protected] (U. Onn), [email protected] (C. Voll). 1 Supported by NSF grant DMS-0901638.

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N2 - We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

AB - We study zeta functions enumerating finite-dimensional irreducible complex linear representations of compact p-adic analytic and of arithmetic groups. Using methods from p-adic integration, we show that the zeta functions associated to certain p-adic analytic pro-p groups satisfy functional equations. We prove a conjecture of Larsen and Lubotzky regarding the abscissa of convergence of arithmetic groups of type A2 defined over number fields, assuming a conjecture of Serre on lattices in semisimple groups of rank greater than 1.

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On representation zeta functions of groups and a conjecture of Larsen-Lubotzky

Abstract

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