Convexes divisibles IV : Structure du bord en dimension 3

Abstract

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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Correspondence to Yves Benoist.

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Benoist, Y. Convexes divisibles IV : Structure du bord en dimension 3. Invent. math. 164, 249–278 (2006). https://doi.org/10.1007/s00222-005-0478-4

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