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Transactions of the American Mathematical Society

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Extendability of Large-Scale Lipschitz Maps

Author: Urs Lang
Journal: Trans. Amer. Math. Soc. 351 (1999), 3975-3988
MSC (1991): Primary 53C20; Secondary 51Kxx, 20F32
Published electronically: February 8, 1999
MathSciNet review: 1698373
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Abstract: Let $X,Y$ be metric spaces, $S$ a subset of $X$, and $f \colon S \to Y$ a large-scale lipschitz map. It is shown that $f$ possesses a large-scale lipschitz extension $\bar f \colon X \to Y$ (with possibly larger constants) if $Y$ is a Gromov hyperbolic geodesic space or the cartesian product of finitely many such spaces. No extension exists, in general, if $Y$ is an infinite-dimensional Hilbert space. A necessary and sufficient condition for the extendability of a lipschitz map $f \colon S \to Y$ is given in the case when $X$ is separable and $Y$ is a proper, convex geodesic space.

References [Enhancements On Off] (What's this?)

  • [B] B. Bowditch: Notes on Gromov's hyperbolicity criterion, in: E. Ghys, A. Haefliger, A. Verjovsky (eds.), Group theory from a geometrical viewpoint, Singapore: World Scientific 1991, pp. 64-167. MR 93h:57002
  • [Bu] H. Busemann: The geometry of geodesics, New York: Academic Press 1955. MR 17:779a
  • [F] B. Farb: The extrinsic geometry of subgroups and the generalized word problem, Proc. London Math. Soc. 68 (1994), 577-593. MR 94m:20073
  • [G1] M. Gromov: Hyperbolic groups, in: S. M. Gersten (ed.), Essays in group theory, MSRI Publ. no. 8, New York: Springer 1987, pp. 75-263. MR 89e:20070
  • [G2] -: Asymptotic invariants of infinite groups, in: G. A. Niblo, M. A. Roller (eds.), Geometric group theory, vol. 2, London Math. Soc. Lecture Note Series no. 182, Cambridge Univ. Press 1993, pp. 1-295. MR 95m:20041
  • [J] J. Jost: Nonpositive curvature. Geometric and analytic aspects, Lectures in Math., ETH Zürich, Basel: Birkhäuser 1997. MR 98g:53070
  • [K] M. D. Kirszbraun: Über die zusammenziehende und Lipschitzsche Transformationen, Fundamenta Math. 22 (1934), 77-108.
  • [LS] U. Lang, V. Schroeder: Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal. (GAFA) 7 (1997), 535-560. MR 98d:53062
  • [M] E. J. McShane: Extension of range of functions, Bull. Amer. Math. Soc. 40 (1934), 837-842.
  • [MoSh] J. W. Morgan, P. B. Shalen: Valuations, trees, and degenerations of hyperbolic structures, I, Ann. of Math. 120 (1984), 401-476. MR 86f:57011
  • [V] N. Varopoulos: Sur la distortion de distances des sous-groupes des groupes de Lie, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), 1025-1026. MR 97d:22010

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Additional Information

Urs Lang
Affiliation: Departement Mathematik, Eidgen Technische Hochschule Zentrum, CH-8092 Zürich, Switzerland

Received by editor(s): August 8, 1997
Published electronically: February 8, 1999
Additional Notes: Supported by the Swiss National Science Foundation.
Article copyright: © Copyright 1999 American Mathematical Society

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