Profinite invariants of arithmetic groups
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...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Cambridge University Press
2020-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509420000432/type/journal_article |
| _version_ | 1827994825313484800 |
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| author | Holger Kammeyer Steffen Kionke Jean Raimbault Roman Sauer |
| author_facet | Holger Kammeyer Steffen Kionke Jean Raimbault Roman Sauer |
| author_sort | Holger Kammeyer |
| collection | DOAJ |
| description | 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.
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| first_indexed | 2024-04-10T04:47:00Z |
| format | Article |
| id | doaj.art-3a5dda75c23e453d94d1b996c073a41e |
| institution | Directory Open Access Journal |
| issn | 2050-5094 |
| language | English |
| last_indexed | 2024-04-10T04:47:00Z |
| publishDate | 2020-01-01 |
| publisher | Cambridge University Press |
| record_format | Article |
| series | Forum of Mathematics, Sigma |
| spelling | doaj.art-3a5dda75c23e453d94d1b996c073a41e2023-03-09T12:34:47ZengCambridge University PressForum of Mathematics, Sigma2050-50942020-01-01810.1017/fms.2020.43Profinite invariants of arithmetic groupsHolger Kammeyer0https://orcid.org/0000-0002-6567-3762Steffen Kionke1https://orcid.org/0000-0002-6447-8527Jean Raimbault2https://orcid.org/0000-0002-1945-2678Roman Sauer3https://orcid.org/0000-0002-2907-6645Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany; E-mail: ,Faculty of Mathematics and Computer Science, FernUniversität in Hagen, 58097 Hagen, Germany; E-mail:Institut de Mathématiques de Toulouse; UMR5219 Université de Toulouse; CNRS UPS IMT, F-31062 Toulouse Cedex 9, France; E-mail:Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany; E-mail: ,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. https://www.cambridge.org/core/product/identifier/S2050509420000432/type/journal_articleprofinite rigidityarithmetic groupsl2-invariants20E1811F75 |
| spellingShingle | Holger Kammeyer Steffen Kionke Jean Raimbault Roman Sauer Profinite invariants of arithmetic groups Forum of Mathematics, Sigma profinite rigidity arithmetic groups l2-invariants 20E18 11F75 |
| title | Profinite invariants of arithmetic groups |
| title_full | Profinite invariants of arithmetic groups |
| title_fullStr | Profinite invariants of arithmetic groups |
| title_full_unstemmed | Profinite invariants of arithmetic groups |
| title_short | Profinite invariants of arithmetic groups |
| title_sort | profinite invariants of arithmetic groups |
| topic | profinite rigidity arithmetic groups l2-invariants 20E18 11F75 |
| url | https://www.cambridge.org/core/product/identifier/S2050509420000432/type/journal_article |
| work_keys_str_mv | AT holgerkammeyer profiniteinvariantsofarithmeticgroups AT steffenkionke profiniteinvariantsofarithmeticgroups AT jeanraimbault profiniteinvariantsofarithmeticgroups AT romansauer profiniteinvariantsofarithmeticgroups |