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Cubulating random groups at density less than $ 1/6$


Authors: Yann Ollivier and Daniel T. Wise
Journal: Trans. Amer. Math. Soc. 363 (2011), 4701-4733
MSC (2010): Primary 20P05, 20F67; Secondary 20F65, 20F05, 20F06
DOI: https://doi.org/10.1090/S0002-9947-2011-05197-4
Published electronically: March 28, 2011
MathSciNet review: 2806688
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Abstract: We prove that random groups at density less than $ \frac16$ act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than $ \frac15$ have codimension-$ 1$ subgroups. In particular, Property $ (T)$ fails to hold at density less than $ \frac15$.

ABSTRACT. Nous prouvons que les groupes aléatoires en densité strictement inférieure à $ \frac16$ agissent librement et cocompactement sur un complexe cubique CAT(0). De plus en densité strictement inférieure à $ \frac15$, ils ont un sous-groupe de codimension $ 1$; en particulier, la propriété $ (T)$ n'est pas vérifiée.


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Additional Information

Yann Ollivier
Affiliation: CNRS, UMPA, École normale supérieure de Lyon, 46, allée d’Italie, 69364 Lyon cedex 7, France
Address at time of publication: CNRS, LRI, Université Paris-Sud, Bat. 490, 91405 Orsay cedex, France
Email: yann.ollivier@umpa.ens-lyon.fr, yann.ollivier@lri.fr

Daniel T. Wise
Affiliation: Department of Mathematics, McGill University, Montreal, Québec, Canada H3A 2K6
Email: wise@math.mcgill.ca

DOI: https://doi.org/10.1090/S0002-9947-2011-05197-4
Keywords: CAT(0) cube complexes, random groups, Property $(T)$
Received by editor(s): August 27, 2008
Received by editor(s) in revised form: August 27, 2009
Published electronically: March 28, 2011
Additional Notes: This research was partially supported by NSERC grant
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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