On the existence of flat planes in spaces of nonpositive curvature

Bridson, Martin R.

doi:10.1090/S0002-9939-1995-1273477-8

On the existence of flat planes in spaces of nonpositive curvature
HTML articles powered by AMS MathViewer

by Martin R. Bridson

Proc. Amer. Math. Soc. 123 (1995), 223-235

DOI: https://doi.org/10.1090/S0002-9939-1995-1273477-8

PDF | Request permission

Abstract:

Let X be a proper 1-connected geodesic metric space which is non-positively curved in the sense that it satisfies Gromov’s ${\text {CAT}}(0)$ condition globally. If X is cocompact, then either it is $\delta$-hyperbolic, for some $\delta > 0$, or else it contains an isometrically embedded copy of the Euclidean plane; these conditions are mutually exclusive. It follows that if the fundamental group of a compact non-positively curved polyhedron K is not word-hyperbolic, then the universal cover of K contains a flat plane.

References

A theorem on triangles in a metric space and some applications

38

Juan M. Alonso, Inégalités isopérimétriques et quasi-isométries, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 12, 761–764 (French, with English summary). MR 1082628
Juan M. Alonso and Martin R. Bridson, Semihyperbolic groups, Proc. London Math. Soc. (3) 70 (1995), no. 1, 56–114. MR 1300841, DOI 10.1112/plms/s3-70.1.56
Werner Ballmann, Singular spaces of nonpositive curvature, Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988) Progr. Math., vol. 83, Birkhäuser Boston, Boston, MA, 1990, pp. 189–201. MR 1086658, DOI 10.1007/978-1-4684-9167-8_{1}0

Manifolds of non-positive curvature

Metric spaces of non-positive curvature

B. H. Bowditch, Minkowskian subspaces of non-positively curved metric spaces, Bull. London Math. Soc. 27 (1995), no. 6, 575–584. MR 1348712, DOI 10.1112/blms/27.6.575
Martin R. Bridson, Geodesics and curvature in metric simplicial complexes, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 373–463. MR 1170372
Herbert Busemann, The geometry of geodesics, Academic Press, Inc., New York, N.Y., 1955. MR 0075623
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Patrick Eberlein, Geodesic flow in certain manifolds without conjugate points, Trans. Amer. Math. Soc. 167 (1972), 151–170. MR 295387, DOI 10.1090/S0002-9947-1972-0295387-4
P. Eberlein and B. O’Neill, Visibility manifolds, Pacific J. Math. 46 (1973), 45–109. MR 336648, DOI 10.2140/pjm.1973.46.45
William Floyd and Walter Parry, The growth of nonpositively curved triangles of groups, Invent. Math. 129 (1997), no. 2, 289–359. MR 1465327, DOI 10.1007/s002220050165
É. Ghys and P. de la Harpe (eds.), Sur les groupes hyperboliques d’après Mikhael Gromov, Progress in Mathematics, vol. 83, Birkhäuser Boston, Inc., Boston, MA, 1990 (French). Papers from the Swiss Seminar on Hyperbolic Groups held in Bern, 1988. MR 1086648, DOI 10.1007/978-1-4684-9167-8
S. M. Gersten and H. B. Short, Small cancellation theory and automatic groups, Invent. Math. 102 (1990), no. 2, 305–334. MR 1074477, DOI 10.1007/BF01233430
M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
Lee Mosher, Geometry of cubulated $3$-manifolds, Topology 34 (1995), no. 4, 789–814. MR 1362788, DOI 10.1016/0040-9383(94)00050-6
A. Yu. Ol′shanskiĭ, Hyperbolicity of groups with subquadratic isoperimetric inequality, Internat. J. Algebra Comput. 1 (1991), no. 3, 281–289. MR 1148230, DOI 10.1142/S0218196791000183
Panagiotis Papasoglu, On the sub-quadratic isoperimetric inequality, Geometric group theory (Columbus, OH, 1992) Ohio State Univ. Math. Res. Inst. Publ., vol. 3, de Gruyter, Berlin, 1995, pp. 149–157. MR 1355109
J. M. Alonso, T. Brady, D. Cooper, V. Ferlini, M. Lustig, M. Mihalik, M. Shapiro, and H. Short, Notes on word hyperbolic groups, Group theory from a geometrical viewpoint (Trieste, 1990) World Sci. Publ., River Edge, NJ, 1991, pp. 3–63. Edited by Short. MR 1170363