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Profinite invariants of arithmetic groups

Published online by Cambridge University Press:  13 November 2020

Holger Kammeyer
Affiliation:
Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131Karlsruhe, Germany; E-mail: [email protected], [email protected]
Steffen Kionke
Affiliation:
Faculty of Mathematics and Computer Science, FernUniversität in Hagen, 58097Hagen, Germany; E-mail: [email protected]
Jean Raimbault
Affiliation:
Institut de Mathématiques de Toulouse; UMR5219 Université de Toulouse; CNRS UPS IMT, F-31062 Toulouse Cedex 9, France; E-mail: [email protected]
Roman Sauer
Affiliation:
Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131Karlsruhe, Germany; E-mail: [email protected], [email protected]

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.

Type
Algebra
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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