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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.

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