149 research outputs found
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
Embedding universal covers of graph manifolds in products of trees
We prove that the universal cover of any graph manifold quasi-isometrically
embeds into a product of three trees. In particular we show that the
Assouad-Nagata dimension of the universal cover of any closed graph manifold is
3, proving a conjecture of Smirnov.Comment: 3 pages, final version - to appear in Proceedings of the AM
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