# 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
Terminology
• 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
Notes from the mini-course Simplicial Nonpositive Curvature by Jacek Świątkowski, on Conference on Geometric Group Theory, Montreal, July 3-14, 2006.
Excercises (list 1, list 2, list 3) from the mini-course on simplicial nonpositive curvature by Jacek Świątkowski, on conference "CAT(0) Cubical and Systolic Complexes", Bedlewo, June 25-29, 2007.

References

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).

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. T. Elsner, P. Przytycki, Square complexes and simplicial nonpositive curvature, Proc. AMS 141 (2013), 2997-3004.
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, Comment. Math. Helv. 88 no. 2 (2013), 353-367.
15. D. Osajda, P. Przytycki, Boundaries of systolic groups, Geometry&Topology 13 (2009), 2807-2880.
16. D. Osajda, J. Świątkowski, On asymptotically hereditarily aspherical groups, (2013), submitted
17. D. Osajda, Combinatorial negative curvature and triangulations of 3-manifolds, submitted.
18. P. Przytycki, Systolic groups acting on complexes with no flats are hyperbolic, Fundamenta Mathematicae 193 (2007), 277-283.
19. P. Przytycki, The fixed point theorem for simplicial nonpositive curvature, Math. Proceedings of the Cambridge Philosophical Society, 144(3) (2008), 683-695.
20. E. Kopczynski, I. Pak, P. Przytycki, Acute triangulations of polyhedra and R^n, Combinatorica 32 no.1 (2012), 85-110.
21. P. Przytycki, J. Świątkowski, Flag-no-square triangulations and Gromov boundaries in dimension 3, Groups, Geometry and Dynamics 3 (2009), 453-468.
22. P. Przytycki, EG for systolic groups, Commentarii Mathematici Helvetici 84 no.1 (2009), 159-169.
23. P. Przytycki, P. Schwer, Systolizing buildings, submitted.
24. J. Świątkowski, Regular path systems and (bi)automatic groups, Geometriae Dedicata 118 (2006), 23-48.
25. J. Świątkowski, Fundamental pro-groups and Gromov boundaries of 7-systolic groups, Journal of the London Mathematical Society 2009 80(3), 649-664.
26. D. Wise, Sixtolic complexes and their fundamental groups, in preparation.
27. P. Zawi¶lak, Trees of manifolds and boundaries of systolic groups, Fundamenta Mathematicae 207 (2010), 71-99.
28. G. Zadnik, Finitely presented subgroups of systolic groups are systolic, submitted.
29. R. Hanlon, E. Martinez-Pedroza, Lifting group actions, equivariant towers and subgroups of non-positively curved groups, submitted.
30. J. Jakus, Weak asymptotic hereditary asphericity for free product and HNN extension of groups, Algebraic&Geometric Topology 13 (2013), 3031-3045.
31. J. Zubik, Asymptotic hereditary asphericity of metric spaces of asymptotic dimension 1, Topology and its Applications, 157(18), (2010), 2815-2818.

Generalizations of simplicial nonpositive curvature

32. V. Chepoi, D. Osajda Dismantlability of weakly systolic complexes and applications, Trans. Amer. Math. Soc. (2014), to appear..
33. S. Hensel, D. Osajda, P. Przytycki, Realization and dismantlability, accepted to Geometry&Topology
34. B. Bresar, J. Chalopin, V. Chepoi, T. Gologranc, D. Osajda Bucolic complexes, Adv. Math. 243 (2013), 127-167.
35. D. Osajda, A combinatorial non-positive cuvature I: weak systolicity, (2013), preprint.