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1 Introduction 1.1 Background and statement of results An (L, C) quasi-isometry is a map Φ : X −→ X ′ between metric spaces such that for all x 1 , x 2 ∈ X we have L −1 d(x 1 , x 2) − C ≤ d(Φ(x 1), Φ(x 2)) ≤ Ld(x 1 , x 2) + C (1) and d(x ′ , Im(Φ)) < C (2) for all x ′ ∈ X ′. Quasi-isometries occur naturally in the study of the geometry of discrete groups… (More)
This is one of a series of papers examining the interplay between differentiation theory for Lipschitz maps, X → V , and bi-Lipschitz nonembeddability, where X is a metric measure space and V is a Banach space. Here, we consider the case V = L 1 , where differentiability fails. We establish another kind of differen-tiability for certain X, including R n and… (More)
We present a new approach to the infinitesimal structure of Lipschitz maps into L 1 , and as an application, we give an alternative proof of the main theorem of [CK06], that the Heisenberg group does not admit a bi-Lipschitz embedding in L 1. The proof uses the metric differentiation theorem of [Pau01] and the cut metric description in [CK06] to reduce the… (More)
According to the classical uniformization theorem, every smooth Riemannian surface Z homeomorphic to the 2-sphere is conformally diffeomorphic to S 2 (the unit sphere in R 3 equipped with the Riemannian metric induced by the ambient Euclidean metric). The availability of a similar uniformization procedure is highly desirable in a nonsmooth setting, in… (More)
If a group acts by uniformly quasi-MM obius homeomor-phisms on a compact Ahlfors n-regular space of topological dimension n such that the induced action on the space of distinct triples is cocompact, then the action is quasi-symmetrically conjugate to an action on the standard n-sphere by MM obius transformations.
— We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap (log n) Ω(1). This is achieved by exhibiting n-point metric spaces of negative type whose L 1 distortion is (log n) Ω(1). Our result is based on quantitative bounds on the rate of degen-eration of Lipschitz maps from the Heisenberg… (More)
We explore the geometry of nonpositively curved spaces with isolated flats, and its consequences for groups that act properly dis-continuously, cocompactly, and isometrically on such spaces. We prove that the geometric boundary of the space is an invariant of the group up to equivariant homeomorphism. We also prove that any such group is relatively… (More)
We call a complement of a union of at least three disjoint (round) open balls in the unit sphere S n a Schottky set. We prove that every quasisymmetric homeomorphism of a Schottky set of spherical measure zero to another Schottky set is the restriction of a Möbius transformation on S n. In the other direction we show that every Schottky set in S 2 of… (More)
We show that a number of diierent notions of dimension coincide for length spaces with curvature bounded above. We then apply this result, showing that if X is a locally compact CAT(0) space with cocompact isometry group, then the dimension of the Tits boundary and the asymptotic cone(s) of X are determined by the maximal dimension of a at in X.
Suppose G is a Gromov hyperbolic group, and ∂ ∞ G is quasisymmetri-cally homeomorphic to an Ahlfors Q-regular metric 2-sphere Z with Ahlfors regular conformal dimension Q. Then G acts discretely, cocompactly, and isometrically on H 3 .