zbMATH — the first resource for mathematics

Summary: Quadric complexes are square complexes satisfying a certain combinatorial nonpositive curvature condition. These complexes generalize $2$-dimensional $$\mathrm{CAT}(0)$$ cube complexes and are a square analog of systolic complexes. We introduce and study the basic properties of these complexes. Using a form of dismantlability for the $$1$$-skeleta of finite quadric complexes, we show that every finite group acting on a quadric complex stabilizes a complete bipartite subgraph of its $1$-skeleton. Finally, we prove that $$\text{C}(4)\mbox{-}\text{T}(4)$$ small cancelation groups act on quadric complexes.
 [1] J. M. Alonso, Inégalités isopérimétriques et quasi-isométries, C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), no. 12, 761-764. · Zbl 0726.57002 [2] R. P. Anstee and M. Farber, On bridged graphs and cop-win graphs, J. Combin. Theory Ser. B 44 (1988), no. 1, 22-28. · Zbl 0654.05049 [3] S. P. Avann, Metric ternary distributive semi-lattices, Proc. Amer. Math. Soc. 12 (1961), no. 3, 407-414. · Zbl 0099.02201 [4] H.-J. Bandelt, Hereditary modular graphs, Combinatorica 8 (1988), no. 2, 149-157. · Zbl 0659.05076 [5] B. Brešar, J. Chalopin, V. Chepoi, T. Gologranc, and D. Osajda, Bucolic complexes, Adv. Math. 243 (2013), 127-167. · Zbl 1314.05238 [6] V. Chepoi, Bridged graphs are cop-win graphs: an algorithmic proof, J. Combin. Theory Ser. B 69 (1997), no. 1, 97-100. · Zbl 0873.05060 [7] V. Chepoi, Graphs of some $$\operatorname{CAT}(0)$$ complexes, Adv. in Appl. Math. 24 (2000), no. 2, 125-179. · Zbl 1019.57001 [8] D. J. Collins and J. Huebschmann, Spherical diagrams and identities among relations, Math. Ann. 261 (1982), no. 2, 155-183. · Zbl 0477.20019 [9] M. Dehn, Papers on group theory and topology, Springer-Verlag, New York, 1987. Translated from the German and with introductions and an appendix by John Stillwell, With an appendix by Otto Schreier. · Zbl 1264.01046 [10] T. Elsner and P. Przytycki, Square complexes and simplicial nonpositive curvature, Proc. Amer. Math. Soc. 141 (2013), no. 9, 2997-3004. · Zbl 1286.20050 [11] V. Gerasimov, Fixed-point-free actions on cubings, Siberian Adv. Math. 8 (1998), no. 3, 26-58. [12] S. M. Gersten, Isoperimetric and isodiametric functions of finite presentations, Geometric group theory: proceedings of the symposium held in Sussex, volume 1, Lond. Math. Soc. Lect. Notes Ser., 181, pp. 79-96, Cambridge University Press, New York, 1991. · Zbl 0829.20054 [13] M. Greendlinger, Dehn’s algorithm for the word problem, Comm. Pure Appl. Math. 13 (1960), no. 1, 67-83. · Zbl 0104.01903 [14] M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., 8, pp. 75-263, Springer, New York, 1987. [15] T. Haettel, Virtually cocompactly cubulated Artin-Tits groups, preprint, 2015, arXiv:1509.08711. [16] F. Haglund, Complexes simpliciaux hyperboliques de grande dimension, preprint, 2003. [17] F. Haglund and D. T. Wise, Special cube complexes, Geom. Funct. Anal. 17 (2008), no. 5, 1551-1620. · Zbl 1155.53025 [18] R. G. Hanlon and E. Martínez-Pedroza, Lifting group actions, equivariant towers and subgroups of non-positively curved groups, Algebr. Geom. Topol. 14 (2014), no. 5, 2783-2808. · Zbl 1335.20045 [19] A. Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. · Zbl 1044.55001 [20] S. Hensel, D. Osajda, and P. Przytycki, Realisation and dismantlability, Geom. Topol. 18 (2014), no. 4, 2079-2126. · Zbl 1320.57022 [21] N. Hoda, Bisimplicial complexes and asphericity, preprint, 2018, arXiv:1804.04630. [22] J. Huang, K. Jankiewicz, and P. Przytycki, Cocompactly cubulated $$2$$-dimensional Artin groups, Comment. Math. Helv. 91 (2016), no. 3, 519-542. · Zbl 1401.20044 [23] J. Huang and D. Osajda, Metric systolicity and two-dimensional Artin groups, preprint, 2017, arXiv:1710.05157. · Zbl 07087154 [24] J. Huebschmann, Cohomology theory of aspherical groups and of small cancellation groups, J. Pure Appl. Algebra 14 (1979), no. 2, 137-143. · Zbl 0396.20021 [25] T. Januszkiewicz and J. Świątkowski, Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci. 104 (2006), no. 1, 1-85. · Zbl 1143.53039 [26] S. Klavžar and H. M. Mulder, Median graphs: characterizations, location theory and related structures, J. Combin. Math. Combin. Comput. 30 (1999), 103-127. · Zbl 0931.05072 [27] R. C. Lyndon and P. E. Schupp, Combinatorial group theory, Classics Math., Springer-Verlag, Berlin, 2001. Reprint of the 1977 edition. · Zbl 0997.20037 [28] J. P. McCammond and D. T. Wise, Fans and ladders in small cancellation theory, Proc. Lond. Math. Soc. (3) 84 (2002), no. 3, 599-644. · Zbl 1022.20012 [29] L. Nebeský, Median graphs, Comment. Math. Univ. Carolin. 12 (1971), no. 2, 317-325. · Zbl 0215.34001 [30] G. A. Niblo and L. D. Reeves, Groups acting on $$\operatorname{CAT}(0)$$ cube complexes, Geom. Topol. 1 (1997), no. 1, 1-7. · Zbl 0887.20016 [31] G. A. Niblo and L. D. Reeves, The geometry of cube complexes and the complexity of their fundamental groups, Topology 37 (1998), no. 3, 621-633. · Zbl 0911.57002 [32] N. Polat, Finite invariant simplices in infinite graphs, Period. Math. Hungar. 27 (1993), no. 2, 125-136. · Zbl 0795.05136 [33] S. J. Pride, On Tits’ conjecture and other questions concerning Artin and generalized Artin groups, Invent. Math. 86 (1986), no. 2, 347-356. · Zbl 0633.20021 [34] M. Roller, Poc sets, median algebras and group actions, Habilitationsschrift, University of Regensburg, 1998. [35] M. Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. Lond. Math. Soc. (3) 71 (1995), no. 3, 585-617. · Zbl 0861.20041 [36] V. P. Soltan and V. D. Chepoi, Conditions for invariance of set diameters under $$d$$-convexification in a graph, Kibernetika (Kiev) 19 (1983), no. 6, 14-18. [37] D. T. Wise, Sixtolic complexes and their fundamental groups, preprint, 2003. [38] D. T. Wise, The structure of groups with a quasiconvex hierarchy, pp. 1-189, 2011 (submitted). [39] D.