Inventiones mathematicae

, Volume 164, Issue 2, pp 249–278 | Cite as

Convexes divisibles IV : Structure du bord en dimension 3

  • Yves Benoist


Divisible convex sets IV: Boundary structure in dimension 3

Let Ω be an indecomposable properly convex open subset of the real projective 3-space which is divisible i.e. for which there exists a torsion free discrete group Γ of projective transformations preserving Ω such that the quotient M := Γ\Ω is compact. We study the structure of M and of ∂Ω, when Ω is not strictly convex:

The union of the properly embedded triangles in Ω projects in M onto an union of finitely many disjoint tori and Klein bottles which induces an atoroidal decomposition of M.

Every non extremal point of ∂Ω is on an edge of a unique properly embedded triangle in Ω and the set of vertices of these triangles is dense in the boundary of Ω (see Figs. 1 to 4).

Moreover, we construct examples of such divisible convex open sets Ω.


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© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Ecole Normale Supérieure-CNRSParisFrance

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