Divisible convex subsets. I.
(Convexes divisibles. I.)

*(French. English summary)*Zbl 1084.37026
Dani, S. G. (ed.) et al., Algebraic groups and arithmetic. Proceedings of the international conference, Mumbai, India, December 17–22, 2001. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research (ISBN 81-7319-618-4/hbk). 339-374 (2004).

A subset \(\Omega\) of the real projective space P\(({\mathbb R}^n)\) is called strictly convex if for each projective line, the intersection with \( \Omega\) is connected and the intersection with \(\partial\Omega\) contains at most two points. Assume that \(\Omega\) is open, convex, and that there exists a hyperplane disjoint from the closure of \(\Omega\). Then \(\Omega\) is called divisible by a discrete subgroup \(\Gamma\) of \(\text{ PGL}_n({\mathbb R})\), if \(\Omega\) is \(\Gamma\)-invariant, and \(\Gamma\backslash \Omega\) is compact.

A typical example is the hyperbolic \(n\)-\(1\)-space, when realized in projective space, which is divisible by each discrete group of isometries for the quadratic form defining \(\Omega\). The author shows that a \(\Gamma\)-divisible convex set \(\Omega\) is strictly convex if and only if \(\partial \Omega\) is \(C^1\). This in turn is equivalent to \(\Gamma\) being hyperbolic, and to the fact that the geodesic flow on \(\Gamma\backslash \Omega\) is Anosov. Further, he shows that under these hypotheses, the flow is topologically mixing. Finally, he gives several rigidity results characterizing hyperbolic space among the divisible strictly convex divisible sets by maximal regularity of the boundary and the existence of a flow invariant density.

For the entire collection see [Zbl 1067.00014].

A typical example is the hyperbolic \(n\)-\(1\)-space, when realized in projective space, which is divisible by each discrete group of isometries for the quadratic form defining \(\Omega\). The author shows that a \(\Gamma\)-divisible convex set \(\Omega\) is strictly convex if and only if \(\partial \Omega\) is \(C^1\). This in turn is equivalent to \(\Gamma\) being hyperbolic, and to the fact that the geodesic flow on \(\Gamma\backslash \Omega\) is Anosov. Further, he shows that under these hypotheses, the flow is topologically mixing. Finally, he gives several rigidity results characterizing hyperbolic space among the divisible strictly convex divisible sets by maximal regularity of the boundary and the existence of a flow invariant density.

For the entire collection see [Zbl 1067.00014].

Reviewer: Joachim Hilgert (Paderborn)

##### MSC:

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |

20F67 | Hyperbolic groups and nonpositively curved groups |

22E40 | Discrete subgroups of Lie groups |

37B05 | Dynamical systems involving transformations and group actions with special properties (minimality, distality, proximality, expansivity, etc.) |