Simplicial Nonpositive Curvature

Simplicial Nonpositive Curvature (SNPC) is a purely combinatorial condition for simplicial complexes that:

• resembles metric nonpositive curvature (NPC)
• does not reduce to NPC, nor to small cancellation
• has many similar consequences as classical NPC
• provides examples different from classical ones, with various new and exotic properties

• systole of a simplicial complex L is the length of the shortest full cycle in L (i.e. a cycle that is the full subcomplex of L)
• a simplicial complex is k-large if it is flag and has systole at least k
  [e.g. 5-large is known as Siebenmann's condition]
• a simplicial complex X is locally k-large if links at every simplex in X is k-large (SNPC = local 6-largeness)
• a simplicial complex is k-systolic if it is simply connected and locally k-large (we abbreviate 6-systolic to systolic)
  [for k>5 we obtain an equivalent definition cutting out the word "locally"]
• a systolic group is a group acting geometrically on a systolic complex

Systolic complexes were introduced by T. Januszkiewicz and J. Świątkowski and independently by F. Haglund in the following papers:

1. T. Januszkiewicz, J. Świątkowski, Simplicial nonpositive curvature, Publ. Math. IHES, 104 (2006), 1-85.
2. F. Haglund, Complexes simpliciaux hyperboliques de grande dimension, preprint, Prepublication Orsay 71 (2003).

   We have learned from V. Chepoi about the theory of bridged graphs, which are precisely the 1-skeleta of systolic complexes. Here are available relevant papers related to bridged graphs.

   Other papers and preprints related to simplicial nonpositive curvature are available below:

3. G. Arzhantseva, M. Bridson, T. Januszkiewicz, I. Leary, A. Minasyan, J. Świątkowski, Infinite groups with fixed point properties, Geometry&Topology 13 (2009), 1229-1263.
4. V. Chepoi, D. Osajda Dismantlability of weakly systolic complexes and applications, submitted.
5. T. Elsner, Flats and the flat torus theorem for systolic spaces, Geometry&Topology 13 (2009), 661-698.
6. T. Elsner, Systolic spaces with isolated flats, submitted.
7. T. Elsner, Isometries of systolic spaces, Fundamenta Mathematicae 204 (2009), 39-55.
8. F. Haglund, J. Świątkowski, Separating quasi-convex subgroups in 7-systolic groups, Groups, Geometry and Dynamics 2 (2008), 223-244.
9. T. Januszkiewicz, J. Świątkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv. 78 (2003), 555-583.
10. T. Januszkiewicz, J. Świątkowski, Filling invariants of systolic complexes and groups, Geometry&Topology 11 (2007), 727-758.
11. T. Januszkiewicz, J. Świątkowski, Nonpositively curved developements of billiards, Journal of Topology 2010; doi: 10.1112/jtopol/jtq001.
12. D. Osajda, Ideal boundary of 7-systolic complexes and groups, Algebraic&Geometric Topology 8 (2008), 81-99.
13. D. Osajda, Connectedness at infinity of systolic complexes and groups, Groups, Geometry and Dynamics 1 (2007), 183-203.
14. D. Osajda Construction of hyperbolic Coxeter groups.
15. P. Przytycki, D. Osajda, Boundaries of systolic groups, Geometry&Topology 13 (2009), 2807-2880.
16. P. Przytycki, Systolic groups acting on complexes with no flats are hyperbolic, Fundamenta Mathematicae 193 (2007), 277-283.
17. P. Przytycki, The fixed point theorem for simplicial nonpositive curvature, Math. Proceedings of the Cambridge Philosophical Society, 144(3) (2008), 683-695.
18. P. Przytycki, J. Świątkowski, Flag-no-square triangulations and Gromov boundaries in dimension 3, Groups, Geometry and Dynamics 3 (2009), 453-468.
19. P. Przytycki, EG for systolic groups, Commentarii Mathematici Helvetici 84 no.1 (2009), 159-169.
20. J. Świątkowski, Regular path systems and (bi)automatic groups, Geometriae Dedicata 118 (2006), 23-48.
21. J. Świątkowski, Fundamental pro-groups and Gromov boundaries of 7-systolic groups, Journal of the London Mathematical Society 2009 80(3), 649-664.
22. D. Wise, Sixtolic complexes and their fundamental groups, in preparation.
23. P. Zawiślak, Trees of manifolds and boundaries of systolic groups, accepted to Fundamenta Mathematicae.