Metriques de Carnot-Caratheodory et Quasiisometries des Espaces Symetriques de rang un

Pansu, Pierre

doi:10.2307/1971484

Cited by 798 publications

(700 citation statements)

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“…In this paper, we continue our investigation of Lipschitz maps of metric spaces into Banach spaces [CK06,CK08a,CK08b], which is motivated by the role of bi-Lipschitz embedding problems in theoretical computer science [LLR95,AR98,Lin02,LN06], earlier developments in the infinitesimal geometry of metric measure spaces [Pan89,Che99], and the geometry of Banach spaces [BL00, Chapters 6-7], [Bou85]. Our main purpose here is to present the details of an approach to Lipschitz maps into L 1 announced in [CK06, Section 1.8], which gives new insight into both embeddability and non-embeddability questions; as a first application, we give a new proof of a (slightly stronger version of) the main result of [CK06].…”

Section: Introductionmentioning

confidence: 99%

Metric differentiation, monotonicity and maps to L 1

2010

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Abstract. This is one of a series of papers on Lipschitz maps from metric spaces to L 1 . Here we present the details of results which were announced in [CK06, Section 1.8]: a new approach to the infinitesimal structure of Lipschitz maps into L 1 , and, as a first application, 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 Pauls [Pau01] and the cut metric description in [CK06] to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group. A quantitative version of this classification argument is used in our forthcoming joint paper with Assaf Naor [CKN].

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Section: Introductionmentioning

confidence: 99%

Metric differentiation, monotonicity and maps to L 1

2010

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“…Hyperbolic groups have very interesting spaces at infinity associated to them which satisfy the doubling property described in the next section. In addition to the references already mentioned, see [19,41,70,71] in this regard. Note that fundamental groups of compact Riemannian manifolds without boundary and with strictly negative sectional curvatures are nonelementary hyperbolic groups.…”

Section: Finitely-generated Groupsmentioning

confidence: 95%

Happy fractals and some aspects of analysis on metric spaces

Semmes¹

2003

Publicacions Matemàtiques

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In this survey we discuss "happy fractals", which are complete metric spaces which are not too big, in the sense of a doubling condition, and for which there is a path between any two points whose length is bounded by a constant times the distance between the two points. We also review some aspects of basic analysis on metric spaces, related to Lipschitz functions, approximations and regularizations of functions, and the notion of "atoms".

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“…But in general this can be done by just understanding how the Carnot metric on the ideal boundary is obtained. See [13] for details. Consider the family of horospheres based at ∞ parametrized by t. Each horosphere has two tangent planes V 2 , V 1 invariant under the action of M .…”

Section: Theorem 1 Let γ ⊂ G Be a Nonelementary Nonparabolic Group Imentioning

confidence: 99%