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

We introduce a family of conditions on a simplicial complex that we call local k-largeness (k≥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{a}tkowski, Jacek},
     title = {Simplicial nonpositive curvature},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {1--85},
     year = {2006},
     publisher = {Springer},
     volume = {104},
     doi = {10.1007/s10240-006-0038-5},
     mrnumber = {2264834},
     language = {en},
     url = {https://www.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  - https://www.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 https://www.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

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

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. | Zbl | MR

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

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

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 | MR

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 | MR

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

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 | MR

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

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 | MR

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

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

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 | MR

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 | MR

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 | MR

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

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

Cité par Sources :