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
Notes from the mini-course Simplicial Nonpositive Curvature
by Jacek Świątkowski, on Conference on Geometric Group Theory, Montreal, July 3-14, 2006.
- 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.
Systolic complexes were introduced by T. Januszkiewicz and J. Świątkowski and independently by F. Haglund in the following papers:
- 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, EG for systolic groups, Commentarii Mathematici Helvetici 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.