Relatively hyperbolic groups: geometry and quasi-isometric invariance

In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct of the arguments, simplified definitions of relative hyperbolicity are obtained. In particular we obtain a new definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic triangle of a central left coset of peripheral subgroup.Comment: 34 pages, Latex; added references, corrected typos, pictures included in the Latex fil

Diophantine approximation on rational quadrics

We compute the Hausdorff dimension of sets of very well approximable vectors on rational quadrics. We use ubiquitous systems and the geometry of locally symmetric spaces. As a byproduct we obtain the Hausdorff dimension of the set of rays with a fixed maximal singular direction, which move away into one end of a locally symmetric space at linear depth, infinitely many times.Comment: 55 pages, 3 figures, to appear in Math. Annalen, revised version : updated references, minor correction

Geometry of infinitely presented small cancellation groups, Rapid Decay and quasi-homomorphisms

We study the geometry of infinitely presented groups satisfying the small cancelation condition C'(1/8), and define a standard decomposition (called the criss-cross decomposition) for the elements of such groups. We use it to prove the Rapid Decay property for groups with the stronger small cancelation property C'(1/10). As a consequence, the Metric Approximation Property holds for the reduced C*-algebra and for the Fourier algebra of such groups. Our method further implies that the kernel of the comparison map between the bounded and the usual group cohomology in degree 2 has a basis of power continuum. The present work can be viewed as a first non-trivial step towards a systematic investigation of direct limits of hyperbolic groups.Comment: 40 pages, 8 figure

Non-linear residually finite groups

We give the first example of a non-linear residually finite 1-related group: .Comment: 5 pages. to appear in J. Algebra, 200

Relatively Hyperbolic Groups with Rapid Decay Property

We prove that a finitely generated group $G$ hyperbolic relative to the collection of finitely generated subgroups H_1,..., H_m has the Rapid Decay property if and only if each H_i, i=1,2,..., m, has the Rapid Decay property.Comment: 10 pages. Accepted in International Mathematics Research Notice