Convex geodesic bicombings and hyperbolicity

详细信息    查看全文

• 作者：Dominic Descombes ; Urs Lang
• 关键词：Geodesic bicombing ; Injective hull ; Tight span ; Hyperbolic group ; Absolute retract ; 53C23 ; 20F65 ; 20F67
• 刊名：Geometriae Dedicata
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：177
• 期：1
• 页码：367-384
• 全文大小：502 KB
• 参考文献：1.Alexander, S.B., Bishop, R.L.: The Hadamard-Cartan theorem in locally convex metric spaces. Enseign. Math. 36, 309-20 (1990)MathSciNet MATH
  2.Ballmann, W.: Lectures on Spaces of Nonpositive Curvature. Birkh?user, Switzerland (1995)
  3.Bestvina, M., Mess, G.: The boundary of negatively curved groups. J. Am. Math. Soc. 4, 469-81 (1991)MathSciNet View Article MATH
  4.Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature. Springer, Berlin (1999)View Article MATH
  5.Buyalo, S., Schroeder, V.: Elements of Asymptotic Geometry. EMS Monographs in Mathematics. EMS Publishing House, Zurich, Switzerland (2007)
  6.Cianciaruso, F., De Pascale, E.: Discovering the algebraic structure on the metric injective envelope of a real Banach space. Topol. Appl. 78, 285-92 (1997)View Article MATH
  7.Dress, A.W.M.: Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces. Adv. Math. 53, 321-02 (1984)MathSciNet View Article MATH
  8.Dugundji, J.: Absolute neighborhood retracts and local connectedness in arbitrary metric spaces. Compositio Math. 13, 229-46 (1958)MathSciNet MATH
  9.Dugundji, J.: Locally equiconnected spaces and absolute neighborhood retracts. Fund. Math. 57, 187-93 (1965)MathSciNet MATH
  10.Foertsch, T., Lytchak, A.: The de Rham decomposition theorem for metric spaces. Geom. Funct. Anal. 18, 120-43 (2008)MathSciNet View Article MATH
  11.Gromov, M.: Hyperbolic groups. In: S.M. Gersten (ed.) Essays in Group Theory, vol. 8, pp. 75-63. Math. Sci. Res. Inst. Publ., Springer, Berlin (1987)
  12.Himmelberg, C.J.: Some theorems on equiconnected and locally equiconnected spaces. Trans. Am. Math. Soc. 115, 43-3 (1965)MathSciNet View Article MATH
  13.Hitzelberger, P., Lytchak, A.: Spaces with many affine functions. Proc. Am. Math. Soc. 135, 2263-271 (2007)MathSciNet View Article MATH
  14.Isbell, J.R.: Six theorems about injective metric spaces. Comment. Math. Helv. 39, 65-6 (1964)MathSciNet View Article
  15.Isbell, J.R.: Injective envelopes of Banach spaces are rigidly attached. Bull. Am. Math. Soc. 70, 727-29 (1964)MathSciNet View Article MATH
  16.Kelley, J.L.: Banach spaces with the extension property. Trans. Am. Math. Soc. 72, 323-26 (1952)MathSciNet View Article
  17.Kleiner, B.: The local structure of length spaces with curvature bounded above. Math. Z. 231, 409-56 (1999)MathSciNet View Article MATH
  18.Lang, U.: Injective hulls of certain discrete metric spaces and groups. J. Topol. Anal. 5, 297-31 (2013)MathSciNet View Article MATH
  19.Nachbin, L.: A theorem of the Hahn-Banach type for linear transformations. Trans. Am. Math. Soc. 68, 28-6 (1950)MathSciNet View Article MATH
  20.Papadopoulos, A.: Metric Spaces, Convexity and Nonpositive Curvature, IRMA Lectures in Mathematics and Theoretical Physics, vol. 6. European Mathematical Society (EMS), Zurich, Switzerland (2005)
  21.Rao, N.V.: The metric injective hulls of normed spaces. Topol. Appl. 46, 13-1 (1992)View Article MATH
  22.Sakai, K.: Geometric Aspects of General Topology, Springer Monographs in Mathematics. Springer, Berlin (2013)View Article
  23.Toruńczyk, H.: Concerning locally homotopy negligible sets and characterization of \(l_2\) -manifolds. Fund. Math. 101, 93-10 (1978)MathSciNet MATH
  
• 作者单位：Dominic Descombes (1)
  Urs Lang (1)
  
  1. Department of Mathematics, ETH Zurich, 8092?, Zurich, Switzerland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9168

文摘

A geodesic bicombing on a metric space selects for every pair of points a geodesic connecting them. We prove existence and uniqueness results for geodesic bicombings satisfying different convexity conditions. In combination with recent work by the second author on injective hulls, this shows that every word hyperbolic group acts geometrically on a proper, finite dimensional space \(X\) with a unique (hence equivariant) convex geodesic bicombing of the strongest type. Furthermore, the Gromov boundary of?\(X\) is a \(Z\)-set in the closure of \(X\), and the latter is a metrizable absolute retract, in analogy with the Bestvina–Mess theorem on the Rips complex.