We introduce a family of conditions on a simplicial complex that we call local -largeness ( 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 -largeness implies non-positive curvature if is sufficiently large. We also show that locally -large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.
@article{PMIHES_2006__104__1_0, author = {Januszkiewicz, Tadeusz and \'Swi\k{a}tkowski, Jacek}, title = {Simplicial nonpositive curvature}, journal = {Publications Math\'ematiques de l'IH\'ES}, pages = {1--85}, publisher = {Springer}, volume = {104}, year = {2006}, doi = {10.1007/s10240-006-0038-5}, mrnumber = {2264834}, language = {en}, url = {http://archive.numdam.org/articles/10.1007/s10240-006-0038-5/} }
TY - JOUR AU - Januszkiewicz, Tadeusz AU - Świątkowski, Jacek TI - Simplicial nonpositive curvature JO - Publications Mathématiques de l'IHÉS PY - 2006 SP - 1 EP - 85 VL - 104 PB - Springer UR - http://archive.numdam.org/articles/10.1007/s10240-006-0038-5/ DO - 10.1007/s10240-006-0038-5 LA - en ID - PMIHES_2006__104__1_0 ER -
%0 Journal Article %A Januszkiewicz, Tadeusz %A Świątkowski, Jacek %T Simplicial nonpositive curvature %J Publications Mathématiques de l'IHÉS %D 2006 %P 1-85 %V 104 %I Springer %U http://archive.numdam.org/articles/10.1007/s10240-006-0038-5/ %R 10.1007/s10240-006-0038-5 %G en %F PMIHES_2006__104__1_0
Januszkiewicz, Tadeusz; Świątkowski, Jacek. Simplicial nonpositive curvature. Publications Mathématiques de l'IHÉS, Tome 104 (2006), pp. 1-85. doi : 10.1007/s10240-006-0038-5. http://archive.numdam.org/articles/10.1007/s10240-006-0038-5/
1. Semihyperbolic groups, Proc. Lond. Math. Soc., III. Ser., 70 (1995), 56-114. | MR | Zbl
and ,2. M. Bestvina, Questions in Geometric Group Theory, http://www.math.utah.edu/∼bestvina.
3. On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc., 127 (1999), no. 7, 2143-2146. | MR | Zbl
,4. Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften 319, Springer, Berlin (1999). | MR | Zbl
and ,5. Hard balls gas and Alexandrov spaces of curvature bounded above, Doc. Math., Extra Vol. ICM II (1998), 289-298. | MR | Zbl
,6. Gluing copies of a 3-dimensional polyhedron to obtain a closed nonpositively curved pseudomanifold, Proc. Amer. Math. Soc., 129 (2001), no. 5, 1493-1498. | MR | Zbl
, , and ,7. Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Amer. J. Math., 115 (1993), no. 5, 929-1009. | MR | Zbl
and ,8. On rigid circuit graphs, Abh. Math. Sem. Univ. Hamb., 25 (1961), 71-76. | MR | Zbl
,9. Word Processing in Groups, Jones and Barlett, Boston, MA (1992). | MR | Zbl
, , , , and ,10. Sur les Groupes Hyperboliques d'apres Mikhael Gromov, Progr. Math., vol. 83, Birkhäuser, Boston, MA (1990). | Zbl
and (eds.),11. Surface subgroups of Coxeter and Artin groups, J. Pure Appl. Algebra, 189 (2004), 135-148. | MR | Zbl
, and ,12. Intersection homology theory, Topology, 19 (1980), no. 2, 135-162. | MR | Zbl
, ,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). | MR
14. Hyperbolic groups, Essays in Group Theory, S. Gersten (ed.), Springer, MSRI Publ. 8 (1987), 75-263. | MR | Zbl
,15. F. Haglund, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71, 2003.
16. Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv., 78 (2003), 555-583. | MR | Zbl
and ,17. Filling invariants in systolic complexes and groups, submitted, 2005.
and ,18. T. Januszkiewicz and J. Świątkowski, Nonpositively curved developments of billiards, preprint, 2006.
19. A model for the universal space for proper actions of a hyperbolic group, New York J. Math., 8 (2002), 1-7. | MR | Zbl
and ,20. I. Leary, A metric Kan-Thurston theorem, in preparation.
21. Every CW-complex is a classifying space for proper bundles, Topology, 40 (2001), 539-550. | MR | Zbl
and ,22. Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, Berlin (1977). | MR | Zbl
and ,23. Regular path systems and (bi)automatic groups, Geom. Dedicata, 118 (2006), 23-48. | MR
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