Conformal structure on the boundary and geodesic flow of a \(\text{CAT}(-1)\)-space. (Structure conforme au bord et flot géodésique d’un \(\text{CAT}(-1)\)-espace.) (French) Zbl 0871.58069

Bourdon, Marc

Enseign. Math., II. Sér. 41, No. 1-2, 63-102 (1995).

The background for the development is the theory of Gromov’s hyperbolic groups [see e.g. M. Gromov, Hyperbolic groups. Essays in group theory, Publ. Math. Sci. Res. Inst. 8, 75-263 (1987; Zbl 0634.20015) and M. Coornaert, T. Delzant and A. Papadopoulos, Géométrie et théorie des groupes. Les groupes hyperbolique de Gromov, (Lect. Notes Math. 1441) (1990; Zbl 0727.20018)]. The author studies actions of hyperbolic groups on the CAT(–1)-spaces, a vast abstract generalization of the hyperbolic spaces defined by means of comparison of geodesic triangles with those in the hyperbolic plane, including, among others, all Riemannian manifolds with sectional curvatures at most \(-1\) and the Gromov polyhedra. The first half of the paper is devoted to a quite detailed exposition of the basic concepts and their properties. Then a distinguished conformal family of metrics is constructed on the boundary of any CAT(–1)-space. Finally, the relations between the geodesic flows given by quasi-convex actions of a hyperbolic group by isometries and the conformal structures on their limit points are studied. The paper is well organized and clearly written.

Reviewer: J.Slovák (Adelaide)

MSC:

37D40	Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25	Geodesic flows in symplectic geometry and contact geometry
53C23	Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords:

hyperbolic spaces; hyperbolic groups; CAT(-1)-spaces; geodesic flow; conformal structures

Citations:

Zbl 0634.20015; Zbl 0727.20018