Abstract
In spaces of nonpositive curvature the existence of isometrically embedded flat (hyper)planes is often granted by apparently weaker conditions on large scales.We show that some such results remain valid for metric spaces with non-unique geodesic segments under suitable convexity assumptions on the distance function along distinguished geodesics. The discussion includes, among other things, the Flat Torus Theorem and Gromov’s hyperbolicity criterion referring to embedded planes. This generalizes results of Bowditch for Busemann spaces.
References
[1] M. Anderson, V. Schroeder, Existence of flats in manifolds of nonpositive curvature, Invent. Math. 85 (1986), 303–315. 10.1007/BF01389092Search in Google Scholar
[2] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, Birkhäuser 1995. 10.1007/978-3-0348-9240-7Search in Google Scholar
[3] J. Behrstock, C. Druµu, M. Sapir, Median structures on asymptotic cones and homomorphisms into mapping class groups, Proc. Lond. Math. Soc. 102 (2011), 503–554. 10.1112/plms/pdq025Search in Google Scholar
[4] B. H. Bowditch, Minkowskian subspaces of non-positively curved metric spaces, Bull. London Math. Soc. 27 (1995), 575– 584. 10.1112/blms/27.6.575Search in Google Scholar
[5] B. H. Bowditch, Some properties of median metric spaces, Groups Geom. Dyn. 10 (2016), 279–317. 10.4171/GGD/350Search in Google Scholar
[6] M. R. Bridson, On the existence of flat planes in spaces of nonpositive curvature, Proc. Amer.Math. Soc. 123 (1995), 223–235. Search in Google Scholar
[7] M. R. Bridson, A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer 1999. 10.1007/978-3-662-12494-9Search in Google Scholar
[8] H. Busemann, Spaces with non-positive curvature, Acta Math. 80 (1948), 259–310. 10.1007/BF02393651Search in Google Scholar
[9] I. Chatterji, C. Druµu, F. Haglund, Kazhdan and Haagerup properties from the median viewpoint, Adv. Math. 225 (2010), 882–921. 10.1016/j.aim.2010.03.012Search in Google Scholar
[10] D. Descombes, Asymptotic rank of spaces with bicombings, arXiv:1510.05393 [math.MG]. Search in Google Scholar
[11] D. Descombes, U. Lang, Convex geodesic bicombings and hyperbolicity, Geom. Dedicata 177 (2015), 367–384. 10.1007/s10711-014-9994-ySearch in Google Scholar
[12] A. W. M. Dress, Trees, tight extensions of metric spaces, and the cohomological dimension of certain groups: a note on combinatorial properties of metric spaces, Adv. in Math. 53 (1984), 321–402. 10.1016/0001-8708(84)90029-XSearch in Google Scholar
[13] P. Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972), 151–170. 10.1090/S0002-9947-1972-0295387-4Search in Google Scholar
[14] J. Eells, Jr., J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109–160. 10.2307/2373037Search in Google Scholar
[15] A. Es-Sahib, H. Heinich, Barycentre canonique pour un espace métrique à courbure négative, in: Séminaire de Probabilités XXXIII, Lecture Notes in Mathematics, 1709, Springer 1999, pp. 355–370. 10.1007/BFb0096526Search in Google Scholar
[16] T. Foertsch, A. Lytchak, V. Schroeder, Nonpositive curvature and the Ptolemy inequality, Int. Math. Res. Not. 2007, Art. ID rnm100, 15 pp. Search in Google Scholar
[17] S. Gähler, G. Murphy, A metric characterization of normed linear spaces, Math. Nachr. 102 (1981), 297–309. 10.1002/mana.19811020125Search in Google Scholar
[18] K. Goebel, M. Koter, A remark on nonexpansive mappings, Canad. Math. Bull. 24 (1981), 113–115. 10.4153/CMB-1981-019-3Search in Google Scholar
[19] D. Gromoll, J. A. Wolf, Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc. 77 (1971), 545–552. 10.1090/S0002-9904-1971-12747-7Search in Google Scholar
[20] M. Gromov, Hyperbolic groups, in: S. M. Gersten (Ed.), Essays in Group Theory,Math. Sci. Res. Inst. Publ., 8, Springer 1987, pp. 75–263. 10.1007/978-1-4613-9586-7_3Search in Google Scholar
[21] J. R. Isbell, Six theorems about injective metric spaces, Comment. Math. Helv. 39 (1964), 65–76. 10.1007/BF02566944Search in Google Scholar
[22] M. Kapovich, B. Leeb, Finsler bordifications of symmetric and certain locally symmetric spaces, arXiv:1505.03593 [math.DG]. Search in Google Scholar
[23] B. Kleiner, The local structure of length spaces with curvature bounded above, Math. Z. 231 (1999), 409–456. 10.1007/PL00004738Search in Google Scholar
[24] U. Lang, Injective hulls of certain discrete metric spaces and groups, J. Topol. Anal. 5 (2013), 297–331. 10.1142/S1793525313500118Search in Google Scholar
[25] H. B. Lawson, Jr., S. T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geom. 7 (1972), 211–228. 10.4310/jdg/1214430828Search in Google Scholar
[26] A. Navas, An L1 ergodic theorem with values in a non-positively curved space via a canonical barycenter map, Ergod. Th. Dynam. Sys. 33 (2013), 609–623. 10.1017/S0143385711001015Search in Google Scholar
[27] A. Papadopoulos, Metric Spaces, Convexity and Nonpositive Curvature, Europ. Math. Soc. 2005. 10.4171/010Search in Google Scholar
[28] A. Preissmann, Quelques propriétés globales des espaces de Riemann, Comment. Math. Helv. 15 (1942–43), 175–216. 10.1007/BF02565638Search in Google Scholar
[29] S. Wenger, The asymptotic rank of metric spaces, Comment. Math. Helv. 86 (2011), 247–275. 10.4171/CMH/223Search in Google Scholar
© 2016 Dominic Descombes and Urs Lang
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.
