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

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.

- T. Januszkiewicz, J. Świątkowski,
*Simplicial nonpositive curvature*, Publ. Math. IHES,**104**(2006), 1-85. - 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:

- G. Arzhantseva, M. Bridson, T. Januszkiewicz, I. Leary, A. Minasyan, J. Świątkowski,
*Infinite groups with fixed point properties*, Geometry&Topology**13**(2009), 1229-1263. - T. Elsner,
*Flats and the flat torus theorem for systolic spaces*, Geometry&Topology**13**(2009), 661-698. - T. Elsner,
*Systolic spaces with isolated flats*, submitted. - T. Elsner,
*Isometries of systolic spaces*, Fundamenta Mathematicae**204**(2009), 39-55. - T. Elsner, P. Przytycki,
*Square complexes and simplicial nonpositive curvature*, Proc. AMS 141 (2013), 2997-3004. - F. Haglund, J. Świątkowski,
*Separating quasi-convex subgroups in 7-systolic groups*, Groups, Geometry and Dynamics**2**(2008), 223-244. - T. Januszkiewicz, J. Świątkowski,
*Hyperbolic Coxeter groups of large dimension*, Comment. Math. Helv.**78**(2003), 555-583. - T. Januszkiewicz, J. Świątkowski,
*Filling invariants of systolic complexes and groups*, Geometry&Topology**11**(2007), 727-758. - T. Januszkiewicz, J. Świątkowski,
*Nonpositively curved developements of billiards*, Journal of Topology 2010; doi: 10.1112/jtopol/jtq001. - D. Osajda,
*Ideal boundary of 7-systolic complexes and groups*, Algebraic&Geometric Topology**8**(2008), 81-99. - D. Osajda,
*Connectedness at infinity of systolic complexes and groups*, Groups, Geometry and Dynamics**1**(2007), 183-203. - D. Osajda,
*Construction of hyperbolic Coxeter groups*, Comment. Math. Helv. 88 no. 2 (2013), 353-367. - D. Osajda, P. Przytycki,
*Boundaries of systolic groups*, Geometry&Topology**13**(2009), 2807-2880. - D. Osajda, J. Świątkowski,
*On asymptotically hereditarily aspherical groups*, (2013), submitted - D. Osajda,
*Combinatorial negative curvature and triangulations of 3-manifolds*, submitted. - P. Przytycki,
*Systolic groups acting on complexes with no flats are hyperbolic*, Fundamenta Mathematicae**193**(2007), 277-283. - P. Przytycki,
*The fixed point theorem for simplicial nonpositive curvature*, Math. Proceedings of the Cambridge Philosophical Society,**144(3)**(2008), 683-695. - E. Kopczynski, I. Pak, P. Przytycki,
*Acute triangulations of polyhedra and R^n*, Combinatorica 32 no.1 (2012), 85-110. - P. Przytycki, J. Świątkowski,
*Flag-no-square triangulations and Gromov boundaries in dimension 3*, Groups, Geometry and Dynamics**3**(2009), 453-468. - P. Przytycki,
, Commentarii Mathematici Helvetici__E__G for systolic groups**84**no.1 (2009), 159-169. - P. Przytycki, P. Schwer,
*Systolizing buildings*, submitted. - J. Świątkowski,
*Regular path systems and (bi)automatic groups*, Geometriae Dedicata**118**(2006), 23-48. - J. Świątkowski,
*Fundamental pro-groups and Gromov boundaries of 7-systolic groups*, Journal of the London Mathematical Society 2009 80(3), 649-664. - D. Wise,
*Sixtolic complexes and their fundamental groups*, in preparation. - P. Zawi¶lak,
*Trees of manifolds and boundaries of systolic groups*, Fundamenta Mathematicae**207**(2010), 71-99. - G. Zadnik,
*Finitely presented subgroups of systolic groups are systolic*, submitted. - R. Hanlon, E. Martinez-Pedroza,
*Lifting group actions, equivariant towers and subgroups of non-positively curved groups*, submitted. - J. Jakus,
*Weak asymptotic hereditary asphericity for free product and HNN extension of groups*, Algebraic&Geometric Topology 13 (2013), 3031-3045. - 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

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

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