Duke Mathematical Journal

Convexes divisibles II

Yves Benoist

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Un cône ouvert proprement convexe C de ℝm est dit divisible si il existe un sousgroupe discret Γ de GL(ℝm) qui préserve C et tel que quotient Γ\C est compact. Nous décrivons l'adhérence de Zariski d'un tel groupe Γ.

Nous montrons que si C n'est ni un produit ni un cône symétrique alors Γ est Zariski dense dans GL(ℝm).


A properly convex open cone in ℝm is called divisible if there exists a discrete subgroup Γ of GLℝm preserving C such that the quotient Γ\C is compact. We describe the Zariski closure of such a group Γ.

We show that if C is divisible but is neither a product nor a symmetric cone, then Γ is Zariski dense in GLℝm.

Article information

Duke Math. J., Volume 120, Number 1 (2003), 97-120.

First available in Project Euclid: 16 April 2004

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 22E40: Discrete subgroups of Lie groups [See also 20Hxx, 32Nxx]
Secondary: 20H15: Other geometric groups, including crystallographic groups [See also 51-XX, especially 51F15, and 82D25] 53A20: Projective differential geometry 57S30: Discontinuous groups of transformations


Benoist, Yves. Convexes divisibles II. Duke Math. J. 120 (2003), no. 1, 97--120. doi:10.1215/S0012-7094-03-12014-1. https://projecteuclid.org/euclid.dmj/1082138626

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See also

  • Yves Benoist. Convexes divisibles. C. R. Acad. Sci. Paris Sér. I Math. Vol. 332, No. 5 (2001), pp. 387-390.