Simplicial nonpositive curvature
Publications Mathématiques de l'IHÉS, Volume 104  (2006), p. 1-85

We introduce a family of conditions on a simplicial complex that we call local $k$-largeness ($k\ge 6$ 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.

@article{PMIHES_2006__104__1_0,
author = {Januszkiewicz, Tadeusz and \'Swi\k atkowski, Jacek},
title = {Simplicial nonpositive curvature},
journal = {Publications Math\'ematiques de l'IH\'ES},
publisher = {Springer},
volume = {104},
year = {2006},
pages = {1-85},
doi = {10.1007/s10240-006-0038-5},
zbl = {pre05117094},
mrnumber = {2264834},
language = {en},
url = {http://www.numdam.org/item/PMIHES_2006__104__1_0}
}

Januszkiewicz, Tadeusz; Świątkowski, Jacek. Simplicial nonpositive curvature. Publications Mathématiques de l'IHÉS, Volume 104 (2006) , pp. 1-85. doi : 10.1007/s10240-006-0038-5. http://www.numdam.org/item/PMIHES_2006__104__1_0/

1. J. Alonso and M. Bridson, Semihyperbolic groups, Proc. Lond. Math. Soc., III. Ser., 70 (1995), 56-114. | MR 1300841 | Zbl 0823.20035

2. M. Bestvina, Questions in Geometric Group Theory, http://www.math.utah.edu/∼bestvina.

3. M. Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc., 127 (1999), no. 7, 2143-2146. | MR 1646316 | Zbl 0928.52007

4. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften 319, Springer, Berlin (1999). | MR 1744486 | Zbl 0988.53001

5. D. Burago, Hard balls gas and Alexandrov spaces of curvature bounded above, Doc. Math., Extra Vol. ICM II (1998), 289-298. | MR 1648079 | 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. | MR 1707510 | 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. | MR 1246182 | Zbl 0804.53056

8. G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamb., 25 (1961), 71-76. | MR 130190 | 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). | MR 1161694 | 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. | MR 2038569 | Zbl 1057.20031

12. M. Goresky, R. Macpherson, Intersection homology theory, Topology, 19 (1980), no. 2, 135-162. | MR 572580 | 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). | MR 1253544

14. M. Gromov, Hyperbolic groups, Essays in Group Theory, S. Gersten (ed.), Springer, MSRI Publ. 8 (1987), 75-263. | MR 919829 | Zbl 0634.20015

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. | MR 1998394 | Zbl 1068.20043

17. T. Januszkiewicz and J. Świątkowski, Filling invariants in systolic complexes and groups, submitted, 2005. | Zbl pre05220928

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. | MR 1887695 | 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. | MR 1838994 | Zbl 0983.55010

22. R. Lyndon and P. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, Berlin (1977). | MR 577064 | Zbl 0368.20023

23. J. Świątkowski, Regular path systems and (bi)automatic groups, Geom. Dedicata, 118 (2006), 23-48. | MR 2239447 | Zbl pre05046837