Relatively hyperbolic groups: geometry and quasi-isometric invariance
نویسنده
چکیده
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 central left coset of peripheral subgroup.
منابع مشابه
Thick metric spaces , relative hyperbolicity , and quasi - isometric rigidity
We study the geometry of non-relatively 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 non-relatively hyperbolic peripheral subgroups is a quasi-isometry invariant...
متن کاملar X iv : m at h / 05 12 59 2 v 4 [ m at h . G T ] 1 J ul 2 00 6 THICK METRIC SPACES , RELATIVE HYPERBOLICITY , AND QUASI - ISOMETRIC RIGIDITY
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 peripheral subgroups is a quasi-isometry invariant. ...
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In this paper we introduce a quasi-isometric invariant class of metric spaces which we call metrically thick. We show that any metrically thick space is not (strongly) relatively hyperbolic with respect to any non-trivial collection of subsets. Further, we show that the property of being (strongly) relatively hyperbolic with thick peripheral subgroups is a quasi-isometry invariant. We show that...
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