Hill, RM;
(2018)
Non-residually Finite Extensions of Arithmetic Groups.
Research in Number Theory
, 5
(2)
10.1007/s40993-018-0140-z.
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Abstract
The aim of the article is to show that there are many finite extensions of arithmetic groups which are not residually finite. Suppose G is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of G has finite extensions which are not residually finite. More precisely, we investigate the group H¯ 2 (Z/n) = lim → Γ H 2 (Γ ,Z/n), where Γ runs through the arithmetic subgroups of G. Elements of H¯ 2 (Z/n) correspond to (equivalence classes of) central extensions of arithmetic groups by Z/n; non-zero elements of H¯ 2 (Z/n) correspond to extensions which are not residually finite. We prove that H¯ 2 (Z/n) contains infinitely many elements of order n, some of which are invariant for the action of the arithmetic completion G[(Q) of G(Q). We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group H¯ 2 (Zl) = lim ←t H¯ 2 (Z/l t ). We show that H¯ 2 (Zl) G[(Q) is isomorphic to Zl c for some positive integer c. When G(R) has no simple components of complex type, we prove that c = b+m, where b is the number of simple components of G(R) and m is the dimension of the centre of a maximal compact subgroup of G(R). In all other cases, we prove upper and lower bounds on c; our lower bound (which we believe is the correct number) is b+m.
| Type: | Article |
|---|---|
| Title: | Non-residually Finite Extensions of Arithmetic Groups |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.1007/s40993-018-0140-z |
| Publisher version: | https://doi.org/10.1007/s40993-018-0140-z |
| Language: | English |
| Additional information: | Copyright © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. |
| Keywords: | cohomology of arithmetic groups, congruence subgroup property, residually finite group |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery.ucl.ac.uk/id/eprint/10059319 |
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