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We introduce a concept of tree-graded metric space and we use it to show quasi-isometry invariance of certain classes of relatively hyperbolic groups, to obtain a characterization of relatively hyperbolic groups in terms of their asymptotic cones, to find geometric properties of Cayley graphs of relatively hyperbolic groups, and to construct the firstâ€¦ (More)

- GOULNARA ARZHANTSEVA, Cornelia Drutu
- 2008

We construct finitely generated groups with arbitrary prescribed Hilbert space compression Î± âˆˆ [0, 1]. For a large class of Banach spaces E (including all uniformly convex Banach spaces), the E-compression of these groups coincides with their Hilbert space compression. Moreover, the groups that we construct have asymptotic dimension at most 3, hence theyâ€¦ (More)

- Cornelia Drutu
- IJAC
- 2002

- Cornelia Drutu
- 2006

In this paper it is proved that relative hyperbolicity is an invariant of quasi-isometry. As a byproduct we provide simplified definitions of relative hyperbolicity in terms of the geometry of a Cayley graph. In particular we obtain a definition very similar to the one of hyperbolicity, relying on the existence for every quasi-geodesic triangle of a centralâ€¦ (More)

We prove that a finitely generated group G hyperbolic relative to the collection of finitely generated subgroups {H1, . . . , Hm} has the Rapid Decay property if and only if each Hi , i = 1, 2, . . . ,m, has the Rapid Decay property.

- Cornelia Drutu
- 2001

We prove that the filling order is quadratic for a large class of solvable groups and asymptotically quadratic for all Q-rank one lattices in semisimple groups of R-rank at least 3. As a byproduct of auxiliary results we give a shorter proof of the theorem on the nondistorsion of horospheres providing also an estimate of a nondistorsion

Divergence functions of a metric space estimate the length of a path connecting two points A, B at distance â‰¤ n avoiding a large enough ball around a third point C. We characterize groups with non-linear divergence functions as groups having cut-points in their asymptotic cones. That property is weaker than the property of having Morse (rank 1)â€¦ (More)

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheralâ€¦ (More)

Preface The main goal of this book is to describe several tools of the quasi-isometric rigidity and to illustrate them by presenting (essentially self-contained) proofs of several fundamental theorems in this area: Gromov's theorem on groups of polynomial growth, Mostow Rigidity Theorem and Schwartz's quasi-isometric rigidity theorem for nonuniform latticesâ€¦ (More)

We prove the existence of a close connection between spaces with measured walls and median metric spaces. We then relate properties (T) and Haagerup (a-T-menability) to actions on median spaces and on spaces with measured walls. This allows us to explore the relationship between the classical properties (T) and Haagerup and their versions using affineâ€¦ (More)