(with M. Bucher-Karlsson) In:

l'Enseign. Math. (2008) pp. 53-55

(with V. Metz and G. Noskov)

We obtain precise descriptions of all horoballs for Hilbert's geometry on simplices and for normed finite-dimensional vector spaces with norm given by some polyhedron. Certain geometrical consequences are deduced and several other applications are pointed out, which concern the dynamics of important classes of nonlinear self-maps of convex cones.

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0) spaces, Gromov hyperbolic spaces, Hilbert geometries, certain pseudoconvex domains, and partially for Thurston’s boundary of Teichmuller spaces. We present several rather general results concerning groups of isometries, as well as the proof of other more specific new theorems, for example concerning the existence of free nonabelian subgroups in CAT(0) geometry, iteration of holomorphic maps, a metric Furstenberg lemma, random walks on groups, noncompactness of automorphism groups of convex cones, and boundary behaviour of Kobayashi’s metric.

(with T. Foertsch)

It is shown that the Hilbert geometry (D,h_D) associated to a bounded convex domain is isometric to a normed vector space if and only if D is an open n-simplex. One further result on the asymptotic geometry of Hilbert's metric is obtained with corollaries for the behavior of geodesics. Finally we prove that every geodesic ray in a Hilbert geometry converges to a point of the boundary.

We prove that almost every path of a random walk on a finitely generated nonamenable group converges in the compactification of the group introduced by W. J. Floyd. In fact, we consider the more general setting of ergodic cocycles of some semigroup of one-Lipschitz maps of a complete metric space with a boundary constructed following Gromov. We obtain in addition that when the Floyd boundary of a finitely generated group is non-trivial, then it is in fact maximal in the sense that it can be identified with the Poisson boundary of the group with reasonable measures. The proof relies on works of Kaimanovich together with visibility properties of Floyd boundaries. Furthermore, we discuss mean proximality of ?? and a conjecture of McMullen. Lastly, related statements about the convergence of certain sequences of points, for example quasigeodesic rays or orbits of one-Lipschitz maps, are obtained.

(with M. Bucher)

We discuss notions of bolic metric spaces introduced by Kasparov and Skandalis. We make some observations which clarify the involved metric axioms and explain all the known basic examples.

(with G. Noskov)

We give some sufficient conditions for Hilbert's metric on convex domains D to be Gromov hyperbolic. The conditions involve an intersecting chords property, which we in turn relate to the Menger curvature of triples of boundary points and, in the case the boundary is smooth, to differential geometric curvature of dD. In particular, the intersecting chords property and hence Gromov hyperbolicity is established for bounded, convex C^2-domains in R^n with non-zero curvature. We also give some necessary conditions for hyperbolicity: the boundary must be of class C^1 and may not conatain a line segment. Furthermore we prove a statement about the asymptotic geometry of the Hilbert metric on arbitrary convex (i.e. non necessarily strictly convex) bounded domains, with an application to maps which do not increase Hilbert distance.