Frédéric Paulin

Pseudogroups of isometries of ℝ and Rips’ theorem on free actions on ℝ-trees

We give a proof of Rips’ theorem that a finitely generated group acting freely on an ℝ-tree is a free product of free abelian groups and surface groups, using methods of dynamical systems and measured foliations.

Outer Automorphisms of Hyperbolic Groups and Small Actions on ℝ-Trees

If Γ is a group, denote by Out(Γ) the group of outer automorphisms of Γ. The definitions of the notions used in this introduction are given in the first section. The main theorem of this paper (section 2) is the following: Theorem.Let Γ be a finitely generated word hyperbolic group. If Out(Γ) is infinite, then there exists an isometric action of Γ on an ℝ-tree with almost cyclic edge stabilizers and without any global fixed point.

Prescribing the behaviour of geodesics in negative curvature

Given a family of (almost) disjoint strictly convex subsets of a complete negatively curved Riemannian manifold M, such as balls, horoballs, tubular neighborhoods of totally geodesic submanifolds, etc, the aim of this paper is to construct geodesic rays or lines in M which have exactly once an exactly prescribed (big enough) penetration in one of them, and otherwise avoid (or do not enter too much in) them. Several applications are given, including a definite improvement of the unclouding problem of [PP1], the prescription of heights of geodesic lines in a finite volume such M, or of spiraling times around a closed geodesic in a closed such M. We also prove that the Hall ray phenomenon described by Hall in special arithmetic situations and by Schmidt-Sheingorn for hyperbolic surfaces is in fact only a negative curvature property.

On the critical exponent of a discrete group of hyperbolic isometries

Abstract Generalizing work of Bishop-Jones in the constant curvature case, we prove that if M is a complete Riemannian manifold with pinched negative curvature and with π1M not virtually nilpotent, then the visual dimension of the set of geodesic rays, starting from a fixed base point, that are recurrent in some compact subset of M, is equal to the critical exponent of π1M. Moreover, the critical exponent of an infinite subgroup H of a given word hyperbolic group equals the visual dimension of the conical limit set of H.

Sur les automorphismes extérieurs des groupes hyperboliques

Abstract We prove that an infinite nilpotent group of outer automorphisms in any word-hyperbolic group fixes projectively an action on an R -tree. In particular, we give short proofs of the theorem that any outer automorphism of a free group has a fixed point in the compactified Culler-Vogtmann Outer Space, and of Scotts conjecture on the rank of the fixed points subgroup of a free group automorphism.

Counting common perpendicular arcs in negative curvature

Let

Groupe modulaire, fractions continues et approximation diophantienne en caractéristique p

D^-

Counting Horoballs and Rational Geodesics

and

Skinning measures in negative curvature and equidistribution of equidistant submanifolds

D^+

Diophantine Approximation in Negatively Curved Manifolds and in the Heisenberg Group

be properly immersed closed locally convex subsets of a Riemannian manifold with pinched negative sectional curvature. Using mixing properties of the geodesic flow, we give an asymptotic formula as