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Divisible convex sets. IV: Boundary structure in dimension 3. (Convexes divisibles. IV: Structure du bord en dimension 3.) (French) Zbl 1107.22006
[Part I in S. G. Dani et al. (eds.), Algebraic groups and arithmetic. 339–374 (2004; Zbl 1084.37026); Part II in Duke Math. J. 120, No. 1, 97–120 (2003; Zbl 1037.22022); Part III in Ann. Sci. Éc. Norm. Supér. (4) 38, No. 5, 793–832 (2005; Zbl 1085.22006).]
A subset $$\Omega$$ in the 3-sphere $$\mathbb{S}^3$$ is called convex, if it intersects each grand circle in a connected set, and properly convex, if in addition there exists a hypersphere not intersecting the closure $$\overline \Omega$$ of $$\Omega$$. Further, such a set is called indecomposable if the cone over $$\Omega$$ in $$\mathbb{R}^4$$ is not the direct sum of two smaller cones. Now suppose that $$\Omega$$ is properly convex and indecomposable. Then $$\Omega$$ is called divisible if there exists a discrete group $$\Gamma$$ of projective linear transformations preserving $$\Omega$$ such that $$M:=\Gamma\backslash \Omega$$ is compact. A 2-simplex $$T$$ in $$\Omega$$ is called a properly embedded triangle, if its boundary is contained in the boundary of $$\Omega$$. The author studies the family $$\mathcal{T}$$ of properly embedded triangles and their images in $$M$$. In particular he shows that $$\Gamma$$ has only finitely many orbits in $$\mathcal{T}$$, that any free abelian subgroup of rank 2 leaves some properly embedded triangle invariant, and that the image of such a triangle in $$M$$ is a finite disjoint union of tori and Klein bottles. As a corollary he finds that $$\Omega$$ is strictly convex if and only if $$\Gamma$$ does not contain a free abelian subgroup of rank 2.

##### MSC:
 22E40 Discrete subgroups of Lie groups 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
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