Simplicial nonpositive curvature

Tadeusz Januszkiewicz; Jacek Świątkowski

Publications Mathématiques de l'IHÉS (2006)

  • Volume: 104, page 1-85
  • ISSN: 0073-8301

Abstract

top
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.

How to cite

top

Januszkiewicz, Tadeusz, and Świątkowski, Jacek. "Simplicial nonpositive curvature." Publications Mathématiques de l'IHÉS 104 (2006): 1-85. <http://eudml.org/doc/104219>.

@article{Januszkiewicz2006,
abstract = {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.},
author = {Januszkiewicz, Tadeusz, Świątkowski, Jacek},
journal = {Publications Mathématiques de l'IHÉS},
keywords = {simplicial complex; -largeness; piecewise euclidean metric; CAT(0) property; Gromov hyperbolicity},
language = {eng},
pages = {1-85},
publisher = {Springer},
title = {Simplicial nonpositive curvature},
url = {http://eudml.org/doc/104219},
volume = {104},
year = {2006},
}

TY - JOUR
AU - Januszkiewicz, Tadeusz
AU - Świątkowski, Jacek
TI - Simplicial nonpositive curvature
JO - Publications Mathématiques de l'IHÉS
PY - 2006
PB - Springer
VL - 104
SP - 1
EP - 85
AB - 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.
LA - eng
KW - simplicial complex; -largeness; piecewise euclidean metric; CAT(0) property; Gromov hyperbolicity
UR - http://eudml.org/doc/104219
ER -

References

top
  1. 1. J. Alonso and M. Bridson, Semihyperbolic groups, Proc. Lond. Math. Soc., III. Ser., 70 (1995), 56–114. Zbl0823.20035MR1300841
  2. 2. M. Bestvina, Questions in Geometric Group Theory, http://www.math.utah.edu/∼bestvina. Zbl1286.57013
  3. 3. M. Bridson, On the semisimplicity of polyhedral isometries, Proc. Amer. Math. Soc., 127 (1999), no. 7, 2143–2146. Zbl0928.52007MR1646316
  4. 4. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Grundlehren der mathematischen Wissenschaften 319, Springer, Berlin (1999). Zbl0988.53001MR1744486
  5. 5. D. Burago, Hard balls gas and Alexandrov spaces of curvature bounded above, Doc. Math., Extra Vol. ICM II (1998), 289–298. Zbl0995.37024MR1648079
  6. 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. Zbl0993.53013MR1707510
  7. 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. Zbl0804.53056MR1246182
  8. 8. G. A. Dirac, On rigid circuit graphs, Abh. Math. Sem. Univ. Hamb., 25 (1961), 71–76. Zbl0098.14703MR130190
  9. 9. D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson and W. Thurston, Word Processing in Groups, Jones and Barlett, Boston, MA (1992). Zbl0764.20017MR1161694
  10. 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). Zbl0731.20025
  11. 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. Zbl1057.20031MR2038569
  12. 12. M. Goresky, R. MacPherson, Intersection homology theory, Topology, 19 (1980), no. 2, 135–162. Zbl0448.55004MR572580
  13. 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). Zbl0841.20039MR1253544
  14. 14. M. Gromov, Hyperbolic groups, Essays in Group Theory, S. Gersten (ed.), Springer, MSRI Publ. 8 (1987), 75–263. Zbl0634.20015MR919829
  15. 15. F. Haglund, Complexes simpliciaux hyperboliques de grande dimension, Prepublication Orsay 71, 2003. 
  16. 16. T. Januszkiewicz and J. Świątkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv., 78 (2003), 555–583. Zbl1068.20043MR1998394
  17. 17. T. Januszkiewicz and J. Świątkowski, Filling invariants in systolic complexes and groups, submitted, 2005. Zbl1188.20043
  18. 18. T. Januszkiewicz and J. Świątkowski, Nonpositively curved developments of billiards, preprint, 2006. Zbl1244.37024
  19. 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. Zbl0990.20027MR1887695
  20. 20. I. Leary, A metric Kan–Thurston theorem, in preparation. 
  21. 21. I. Leary and B. Nucinkis, Every CW-complex is a classifying space for proper bundles, Topology, 40 (2001), 539–550. Zbl0983.55010MR1838994
  22. 22. R. Lyndon and P. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 89, Springer, Berlin (1977). Zbl0368.20023MR577064
  23. 23. J. Świątkowski, Regular path systems and (bi)automatic groups, Geom. Dedicata, 118 (2006), 23–48. Zbl1165.20036MR2239447

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.