Comptes Rendus
Differential Topology
Isotropic nonarchimedean S-arithmetic groups are not left orderable
Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 417-420.

If OS is the ring of S-integers of an algebraic number field F, and OS has infinitely many units, we show that no finite-index subgroup of SL(2,OS) is left orderable. (Equivalently, these subgroups have no nontrivial orientation-preserving actions on the real line.) This implies that if G is an isotropic F-simple algebraic group over an algebraic number field F, then no nonarchimedean S-arithmetic subgroup of G is left orderable. Our proofs are based on the fact, proved by D. Carter, G. Keller, and E. Paige, that every element of SL(2,OS) is a product of a bounded number of elementary matrices.

Si OS est l'anneau des S-entiers d'un corps de nombres F, et OS a une infinité d'unités, nous prouvons qu'aucun sous-groupe d'indice fini de SL(2,OS) n'est ordonnable à gauche. (En d'autres termes, les sous-groupes d'indice fini de SL(2,OS) ne possèdent pas d'action non triviale sur la droite réelle respectant l'orientation.) Cela implique que si G est un groupe algébrique F-simple isotrope, défini sur un corps de nombres F, alors aucun sous-groupe S-arithmétique non-archimédien de G n'est ordonnable à gauche. La démonstration est fondée sur le fait, dû à D. Carter, G. Keller, et E. Paige, que chaque élément de SL(2,OS) est le produit d'un nombre borné de matrices élémentaires.

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DOI: 10.1016/j.crma.2004.07.015
Lucy Lifschitz 1; Dave Witte Morris 2, 3

1 Department of Mathematics, University of Oklahoma, Norman, OK 73019, USA
2 Department of Mathematics and Computer Science, University of Lethbridge, Lethbridge, Alberta T1K 3M4, Canada
3 Department of Mathematics, Oklahoma State University, Stillwater, OK 74078, USA
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Lucy Lifschitz; Dave Witte Morris. Isotropic nonarchimedean S-arithmetic groups are not left orderable. Comptes Rendus. Mathématique, Volume 339 (2004) no. 6, pp. 417-420. doi : 10.1016/j.crma.2004.07.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.07.015/

[1] D. Carter, G. Keller, E. Paige, Bounded expressions in SL(n,A), preprint

[2] G. Cooke; P.J. Weinberger On the construction of division chains in algebraic number rings, with applications to SL2, Commun. Algebra, Volume 3 (1975), pp. 481-524

[3] M.K. Da̧bkowski; J.H. Przytycki; A.A. Togha Non-left-orderable 3-manifold groups (preprint) | arXiv

[4] É. Ghys Actions de réseaux sur le cercle, Invent. Math., Volume 137 (1999) no. 1, pp. 199-231

[5] É. Ghys Groups acting on the circle, Ens. Math., Volume 47 (2001) no. 3/4, pp. 329-407

[6] B. Liehl Beschränkte Wortlänge in SL2, Math. Z., Volume 186 (1984) no. 4, pp. 509-524

[7] V.K. Murty Bounded and finite generation of arithmetic groups (K. Dilcher, ed.), Number Theory, Halifax, NS, CMS Conf. Proc., vol. 15, Amer. Math. Soc., Providence, RI, 1994, pp. 249-261

[8] M.S. Raghunathan On the congruence subgroup problem, II, Invent. Math., Volume 85 (1986), pp. 73-117

[9] D. Witte Arithmetic groups of higher Q-rank cannot act on 1-manifolds, P. Am. Math. Soc., Volume 122 (1994) no. 2, pp. 333-340

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