François Gautero

Relative hyperbolization of (one-ended hyperbolic)-by-cyclic groups

We show that the semi-direct product of a one-ended torsion-free word-hyperbolic group G with Z is a hyperbolic group relative to certain canonical subgroups of G on which the automorphism coming from the action of Z on G acts periodically or with linear growth.

Patterns and minimal dynamics for graph maps

We study the rigidity problem for periodic orbits of (continuous) graph maps belonging to the same homotopy equivalence class. Since the underlying spaces are not necessarily homeomorphic, we define a new notion of pattern which enables us to compare periodic orbits of self-maps of homotopy-equivalent spaces. This definition unifies the known notions of pattern for other spaces. The two main results of the paper are as follows: given a free group endomorphism, we study the persistence under homotopy of the periodic orbits of its topological representatives, and in the irreducible case, we prove the minimality (within the homotopy class) of the set of periodic orbits of its efficient representatives.

Dynamical 2-Complexes

We introduce the class of dynamical 2-complexes. These complexes particularly allow the obtaining of a topological representation of any free group automorphism. A dynamical 2-complex can be roughly defined as a special polyhedron or standard 2-complex equipped with an orientation on its 1-cells satisfying two simple combinatorial properties. These orientations allow us to define non-singular semi-flows on the complex. The relationship with the free group automorphisms is done via a cohomological criterion to foliate the complex by compact graphs.

A non-trivial example of a free-by-free group with the Haagerup property

The aim of this note is to prove that the group of Formanek-Procesi acts properly isometrically on a finite dimensional CAT(0) cube complex. This gives a first example of a non-linear semidirect product between two non abelian free groups which satisfies the Haagerup property.

Combinatorial suspension for disc homeomorphisms

For a punctured disc homeomorphism given combinatorially, we give an algorithmic construction of the suspension flow in the corresponding mapping-torus M3. In particular, one computes explicitly the embedding in the 3-manifold M3 of any finite collection of periodic orbits for this flow. All these orbits are realized as closed braids carried by a branched surface (or template), which we construct in the algorithm. Our construction gives a combinatorial proof of the fact that the periodic orbits of such a suspension flow are carried by a same branched surface.

Cohomological characterization of relative hyperbolicity and combination theorem

We give a cohomological characterization of Gromov relative hyperbolicity. As an application we prove a converse to the combination theorem for graphs of relatively hyperbolic groups given in [9]. We build upon and follow the ideas of the work of S. M. Gersten [11] about the same topics in the classical Gromov hyperbolic setting.

Cross sections to semi-flows on 2-complexes

A dynamical 2-complex is a 2-complex equipped with a set of combinatorial properties which allow to define non-singular semi-flows on the complex. After giving a combinatorial characterization of the dynamical 2-complexes which define hyperbolic attractors when embedded in compact 3-manifolds, one gives an effective criterion for the existence of cross-sections to the semi-flows on these 2-complexes. In the embedded case, this gives an effective criterion of existence of cross-sections to the associated hyperbolic attractors. We present a similar criterion for boundary-tangent flows on compact 3-manifolds which are constructed by means of our dynamical 2-complexes.

Dynamical n-complexes

Abstract Using the special n -polyhedra of Matveev 2, we give a generalization to any dimension n ≥2 of the dynamical 2-complexes introduced in the authors thesis 1. We are particularly interested in codimension 1-foliations with compact leaves of these complexes. One is lead to define a particular class of homotopy equivalences between special polyhedra, which are a generalization, in high dimension, of the Whitehead moves on graphs and of the Matveev moves on special 2-polyhedra 3.