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Random groups and nonarchimedean lattices

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  16 October 2014

SYLVAIN BARRÉ  and
MIKAËL PICHOT
SYLVAIN BARRÉ
Affiliation:
Université de Bretagne Sud, Université Européenne de Bretagne, France; sylvain.barre@univ.ubs.fr
MIKAËL PICHOT
Affiliation:
Department of Mathematics & Statistics, McGill University, Montréal, Québec,Canada H3A 2K6; pichot@math.mcgill.ca

Abstract

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We consider models of random groups in which the typical group is of intermediate rank (in particular, it is not hyperbolic). These models are parallel to Gromov’s well-known constructions, and include for example a ‘density model’ for groups of intermediate rank. The main novelty is the higher rank nature of the random groups. They are randomizations of certain families of lattices in algebraic groups (of rank 2) over local fields.

MSC classification

Secondary: 20F65: Geometric group theory 20E42: Groups with a $BN$-pair; buildings 20E18: Limits, profinite groups
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e26
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/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2014

References

Ballmann, W. and Swiatkowski, J., ‘On L 2-cohomology and property (T) for automorphism groups of polyhedral cell complexes’, Geom. Funct. Anal. 7 (4) (1997), 615–645.CrossRefGoogle Scholar
Barré, S. and Pichot, M., ‘Intermediate rank and property RD’, arXiv:0710.1514.Google Scholar
Barré, S. and Pichot, M., ‘Removing chambers in Bruhat–Tits buildings’, Israel J. Math. (accepted), arXiv:1003.4614.Google Scholar
Barré, S. and Pichot, M., ‘An exotic group with the Haagerup property’, arXiv:1205.1128.Google Scholar
Bekka, B. and de la Harpe, P., Valette, Alain Kazhdan’s Property (T), New Mathematical Monographs, 11 , (Cambridge University Press, Cambridge, 2008).CrossRefGoogle Scholar
Borel, A., Cohomologie de certains groupes discrets et laplaciens p-adiques [d’après H. Garland], Séminaire Bourbaki, exposé no 437, 1973.Google Scholar
Cartwright, D. I. and Steger, T., ‘A family of à n -groups’, Israel J. Math. 103 (1998), 125–140.CrossRefGoogle Scholar
Deligne, P., ‘Extensions centrales non résiduellement finies de groupes arithmétiques’, C. R. Acad. Sci. Paris (1978).Google Scholar
Feit, W. and Higman, G., ‘The nonexistence of certain generalized polygons’, J. Algebra 1 (1964), 114–131.CrossRefGoogle Scholar
Garland, H., ‘p-adic curvature and the cohomology of discrete subgroups of p-adic groups’, Ann. of Math. t. 97 (1973), 375–423.CrossRefGoogle Scholar
Ghys, Étienne Groupes aléatoires (d’après Misha Gromov, $\dots$ ) Astérisque No. 294 (2004), viii, 173–204.Google Scholar
Gromov, M., ‘Hyperbolic manifolds, groups and actions’, inRiemann Surfaces and Related Topics: Riemann Surfaces and Related Topics, Proceedings of the 1978 Stony Brook Conference (1978).Google Scholar
Gromov, M., ‘Infinite groups as geometric objects’, inProc. Int. Congress Math. Warsaw 1983, Vol. 1 (1984), 385–392.Google Scholar
Gromov, M., ‘Hyperbolic groups’, inEssays in Group Theory, Mathematical Sciences Research Institute Publications, 8 (Springer, New York, 1987), 75–263.CrossRefGoogle Scholar
Gromov, M., Geometric Group Theory, Vol. 2, Proceedings of the Symposium Held at Sussex University, Sussex, July 1991, (eds Niblo, G. A. and Roller, M. A.), London Mathematical Society Lecture Note Series, 182 (Cambridge University Press, Cambridge, 1993).Google Scholar
Gromov, M., Spaces and questions (Tel Aviv, 1999), Geom. Funct. Anal. Special Volume (2000), Part I, 118–161.CrossRefGoogle Scholar
Gromov, M., Random walk in random groups, preprint IHES (2001), GAFA, Geom. Funct. Anal., Vol. 13 (2003).CrossRefGoogle Scholar
Lubotzky, A., Samuels, B. and Vishne, U., ‘Explicit constructions of Ramanujan complexes of type à d ’, European J. Combin. 26 (6) (2005), 965–993 (English summary).CrossRefGoogle Scholar
Margulis, G. A., ‘Discrete subgroups of semisimple Lie groups’, inErgebnisse der Mathematik und ihrer Grenzgebiete (3), [Results in Mathematics and Related Areas (3)] 17 (Springer-Verlag, Berlin, 1991).Google Scholar
Mostow, G. D., ‘Strong rigidity of locally symmetric spaces’, Annals of Mathematics Studies, 78 (Princeton University Press, Princeton, NJ, 1973), University of Tokyo Press, Tokyo.Google Scholar
Ollivier, Y., ‘A January 2005 invitation to random groups’, inEnsaios Matemticos [Mathematical Surveys], 10 (Sociedade Brasileira de Matematica, Rio de Janeiro, 2005), (See also ‘January 2010 random groups updates’).Google Scholar
Ollivier, Y., ‘Sharp phase transition theorems for hyperbolicity of random groups’, GAFA, Geom. Funct. Anal. 14 (3) (2004), 595–679.CrossRefGoogle Scholar
Pansu, P., ‘Formules de Matsushima, de Garland et proprit (T) pour des groupes agissant sur des espaces symtriques ou des immeubles’, Bull. Soc. Math. France 126 (1998), 107–139.CrossRefGoogle Scholar
Wang, M. T., ‘A fixed point theorem of discrete group actions on Riemannian manifolds’, J. Differential Geom. 50 (2) (1998), 249–267.CrossRefGoogle Scholar
Żuk, A., ‘La propriété (T) pour les groupes agissant sur les polyèdres’, C. R. Acad. Sci. Paris, t. 323 (1996), 453–458.Google Scholar