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Profinite invariants of arithmetic groups

Part of: Discontinuous groups and automorphic forms Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  13 November 2020

Holger Kammeyer Open the ORCID record for Holger Kammeyer [Opens in a new window] ,
Steffen Kionke Open the ORCID record for Steffen Kionke [Opens in a new window] ,
Jean Raimbault Open the ORCID record for Jean Raimbault [Opens in a new window]  and
Roman Sauer Open the ORCID record for Roman Sauer [Opens in a new window]
Holger Kammeyer
Affiliation:
Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany; E-mail: holger.kammeyer@kit.edu, roman.sauer@kit.edu
Steffen Kionke
Affiliation:
Faculty of Mathematics and Computer Science, FernUniversität in Hagen, 58097 Hagen, Germany; E-mail: steffen.kionke@fernuni-hagen.de
Jean Raimbault
Affiliation:
Institut de Mathématiques de Toulouse; UMR5219 Université de Toulouse; CNRS UPS IMT, F-31062 Toulouse Cedex 9, France; E-mail: Jean.Raimbault@math.univ-toulouse.fr
Roman Sauer
Affiliation:
Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany; E-mail: holger.kammeyer@kit.edu, roman.sauer@kit.edu

Abstract

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We prove that the sign of the Euler characteristic of arithmetic groups with the congruence subgroup property is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type F. Our methods imply similar results for $\ell^2$-torsion as well as a strong profiniteness statement for Novikov–Shubin invariants.

Keywords

profinite rigidityarithmetic groupsl2-invariants

MSC classification

Primary: 20E18: Limits, profinite groups
Secondary: 11F75: Cohomology of arithmetic groups

Information

Type
Algebra
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e54
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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