Projections and relative hyperbolicity

We give an alternative definition of relative hyperbolicity based on properties of closest-point projections on peripheral subgroups. We also derive a distance formula for relatively hyperbolic groups, similar to the one for mapping class groups.Comment: The previous version has been split, the present version is a revision of the first part of the old version. The second part is now called "Tree-graded asymptotic cones

Tree-graded asymptotic cones

We study the bilipschitz equivalence type of tree-graded spaces, showing that asymptotic cones of relatively hyperbolic groups (resp. asymptotic cones of groups containing a cut-point) only depend on the bilipschitz equivalence types of the pieces in the standard (resp. minimal) tree-graded structure. In particular, the asymptotic cones of many relatively hyperbolic groups do not depend on the scaling factor. We also describe the asymptotic cones as above "explicitly". Part of these results were obtained independently and simultaneously by D. Osin and M. Sapir.Comment: Part of http://arxiv.org/abs/1010.4552v3, that has been split. To appear in Groups, Geometry and Dynamic

Separable and tree-like asymptotic cones of groups

Using methods from nonstandard analysis, we will discuss which metric spaces can be realized as asymptotic cones. Applying the results we will find in the context of groups, we will prove that a group with "a few" separable asymptotic cones is virtually nilpotent, and we will classify the real trees appearing as asymptotic cones of (not necessarily hyperbolic) groups.Comment: The hypothesis of Theorem 1.2 had to be strengthene