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. T. Elsner, Flats and the flat torus theorem for systolic spaces, Geometry&Topology 13 (2009), 661-698.
5. T. Elsner, Systolic spaces with isolated flats, submitted.
6. T. Elsner, Isometries of systolic spaces, Fundamenta Mathematicae 204 (2009), 39-55.
7. F. Haglund, J. Świątkowski, Separating quasi-convex subgroups in 7-systolic groups, Groups, Geometry and Dynamics 2 (2008), 223-244.
8. T. Januszkiewicz, J. Świątkowski, Hyperbolic Coxeter groups of large dimension, Comment. Math. Helv. 78 (2003), 555-583.
9. T. Januszkiewicz, J. Świątkowski, Filling invariants of systolic complexes and groups, Geometry&Topology 11 (2007), 727-758.
10. T. Januszkiewicz, J. Świątkowski, Nonpositively curved developements of billiards, submitted.
11. D. Osajda, Ideal boundary of 7-systolic complexes and groups, Algebraic&Geometric Topology 8 (2008), 81-99.
12. D. Osajda, Connectedness at infinity of systolic complexes and groups, Groups, Geometry and Dynamics 1 (2007), 183-203.
13. P. Przytycki, D. Osajda, Boundaries of systolic groups, submitted.
14. P. Przytycki, Systolic groups acting on complexes with no flats are hyperbolic, Fundamenta Mathematicae 193 (2007), 277-283.
15. P. Przytycki, The fixed point theorem for simplicial nonpositive curvature, Math. Proceedings of the Cambridge Philosophical Society, 144(3) (2008), 683-695.
16. P. Przytycki, J. Świątkowski, Flag-no-square triangulations and Gromov boundaries in dimension 3, submitted.
17. P. Przytycki, EG for systolic groups, submitted.
18. J. Świątkowski, Regular path systems and (bi)automatic groups, Geometriae Dedicata 118 (2006), 23-48.
19. J. Świątkowski, Fundamental pro-groups and Gromov boundaries of 7-systolic groups, preprint.
20. D. Wise, Sixtolic complexes and their fundamental groups, in preparation.
21. P. Zawiślak, Gromov boundary of 7-systolic 3-pseudomanifolds, PhD dissertation in preparation.