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Simplicial nonpositive curvature. (English) Zbl 1143.53039
Summary: We introduce a family of conditions on a simplicial complex that we call local \(k\)-largeness (\(k\geq 6\) is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that \(k\)-largeness implies non-positive curvature if \(k\) is sufficiently large. We also show that locally \(k\)-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.

MSC:
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C22 Geodesics in global differential geometry
53C20 Global Riemannian geometry, including pinching
53C45 Global surface theory (convex surfaces à la A. D. Aleksandrov)
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References:
[1] J. Alonso and M. Bridson, Semihyperbolic groups, Proc. Lond. Math. Soc., III. Ser., 70 (1995), 56–114. · Zbl 0823.20035
[2] M. Bestvina, Questions in Geometric Group Theory, http://www.math.utah.edu/estvina. · Zbl 1286.57013
[3] M. Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc., 127 (1999), no. 7, 2143–2146. · Zbl 0928.52007
[4] M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften 319, Springer, Berlin (1999).
[5] D. Burago, Hard balls gas and Alexandrov spaces of curvature bounded above, Doc. Math., Extra Vol. ICM II (1998), 289–298. · Zbl 0995.37024
[6] D. Burago, S. Ferleger, B. Kleiner and A. Kononenko, Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold, Proc. Amer. Math. Soc., 129 (2001), no. 5, 1493–1498. · Zbl 0993.53013
[7] R. Charney and M. Davis, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer. J. Math., 115 (1993), no. 5, 929–1009. · Zbl 0804.53056
[8] G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamb., 25 (1961), 71–76. · Zbl 0098.14703
[9] D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson and W. Thurston, Word Processing in Groups, Jones and Barlett, Boston, MA (1992). · Zbl 0764.20017
[10] E. Ghys and P. de la Harpe (eds.), Sur les Groupes Hyperboliques d’apres Mikhael Gromov, Progr. Math., vol. 83, Birkhäuser, Boston, MA (1990). · Zbl 0731.20025
[11] C. McA. Gordon, D. D. Long and A. W. Reid, Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra, 189 (2004), 135–148. · Zbl 1057.20031
[12] M. Goresky, R. MacPherson, Intersection homology theory, Topology, 19 (1980), no. 2, 135–162. · Zbl 0448.55004
[13] M. Gromov, Asymptotic invariants of infinite groups, Geometric Group Theory, G. Niblo and M. Roller (eds.), LMS Lecture Notes Series 182, vol. 2, Cambridge Univ. Press (1993). · Zbl 0841.20039
[14] M. Gromov, Hyperbolic groups, Essays in Group Theory, S. Gersten (ed.), Springer, MSRI Publ. 8 (1987), 75–263.
[15] F. Haglund, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71, 2003.
[16] T. Januszkiewicz and J. Świątkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv., 78 (2003), 555–583. · Zbl 1068.20043
[17] T. Januszkiewicz and J. Świątkowski, Filling invariants in systolic complexes and groups, submitted, 2005.
[18] T. Januszkiewicz and J. Świątkowski, Nonpositively curved developments of billiards, preprint, 2006.
[19] D. Meintrup and T. Schick, A model for the universal space for proper actions of a hyperbolic group, New York J. Math., 8 (2002), 1–7. · Zbl 0990.20027
[20] I. Leary, A metric Kan–Thurston theorem, in preparation.
[21] I. Leary and B. Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology, 40 (2001), 539–550. · Zbl 0983.55010
[22] R. Lyndon and P. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, Berlin (1977). · Zbl 0368.20023
[23] J. Świątkowski, Regular path systems and (bi)automatic groups, Geom. Dedicata, 118 (2006), 23–48. · Zbl 1165.20036
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