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Metric systolicity and two-dimensional Artin groups. (English) Zbl 07087154
Summary: We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying background for further studies concerning their quasi-isometric rigidity.

MSC:
20F65 Geometric group theory
20F36 Braid groups; Artin groups
20F67 Hyperbolic groups and nonpositively curved groups
20F06 Cancellation theory of groups; application of van Kampen diagrams
20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects)
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References:
[1] Appel, K.I.: On Artin groups and Coxeter groups of large type. In: Contributions to Group Theory, Contemporary Mathematics, vol. 33, pp. 50-78. American Mathematical Society, Providence, RI (1984) · Zbl 0576.20021
[2] Appel, KI; Schupp, PE, Artin groups and infinite Coxeter groups, Invent. Math., 72, 201-220, (1983) · Zbl 0536.20019
[3] Brady, N., Crisp, J.: Two-dimensional Artin groups with CAT(0) dimension three. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), pp. 185-214 (2002) · Zbl 1070.20043
[4] Bell, RW, Three-dimensional FC Artin groups are CAT(0), Geom. Dedic., 113, 21-53, (2005) · Zbl 1134.20038
[5] Bestvina, M., Non-positively curved aspects of Artin groups of finite type, Geom. Topol., 3, 269-302, (1999) · Zbl 0998.20034
[6] Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319. Springer, Berlin (1999)
[7] Brady, T.; McCammond, JP, Three-generator Artin groups of large type are biautomatic, J. Pure Appl. Algebra, 151, 1-9, (2000) · Zbl 1004.20023
[8] Brady, T.; McCammond, JP, Braids, posets and orthoschemes, Algebraic Geom. Topol., 10, 2277-2314, (2010) · Zbl 1205.05246
[9] Brady, T., Artin groups of finite type with three generators, Mich. Math. J., 47, 313-324, (2000) · Zbl 0996.20022
[10] Brieskorn, E.; Saito, K., Artin-gruppen und Coxeter-gruppen, Invent. Math., 17, 245-271, (1972) · Zbl 0243.20037
[11] Charney, R.; Crisp, J., Automorphism groups of some affine and finite type Artin groups, Math. Res. Lett., 12, 321-333, (2005) · Zbl 1077.20055
[12] Charney, R., Davis, M.W.: Finite K(\(\pi \); 1)s for Artin groups. In: Prospects in Topology (Princeton, NJ, 1995), pp. 110-124 (1994) · Zbl 0930.55006
[13] Charney, R.; Davis, MW, The K(\(\pi \); 1)-problem for hyperplane complements associated to infinite re ection groups, J. Am. Math. Soc., 8, 597-627, (1995) · Zbl 0833.51006
[14] Crisp, J.; Godelle, E.; Wiest, B., The conjugacy problem in subgroups of right-angled Artin groups, J. Topol., 2, 442-460, (2009) · Zbl 1181.20030
[15] Charney, R., Artin groups of finite type are biautomatic, Math. Ann., 292, 671-683, (1992) · Zbl 0736.57001
[16] Charney, R., Geodesic automation and growth functions for Artin groups of finite type, Math. Ann., 301, 307-324, (1995) · Zbl 0813.20042
[17] Chepoi, V., Graphs of some CAT(0) complexes, Adv. Appl. Math., 24, 125-179, (2000) · Zbl 1019.57001
[18] Chermak, A., Locally non-spherical Artin groups, J. Algebra, 200, 56-98, (1998) · Zbl 0901.20025
[19] Chepoi, V.; Osajda, D., Dismantlability of weakly systolic complexes and applications, Trans. Am. Math. Soc., 367, 1247-1272, (2015) · Zbl 1376.20047
[20] Conner, GR, Discreteness properties of translation numbers in solvable groups, J. Group Theory, 3, 77-94, (2000) · Zbl 0956.20039
[21] Charney, R.; Paris, L., Convexity of parabolic subgroups in Artin groups, Bull. Lond. Math. Soc., 46, 1248-1255, (2014) · Zbl 1308.20037
[22] Crisp, J., Automorphisms and abstract commensurators of 2-dimensional Artin groups, Geom. Topol., 9, 1381-1441, (2005) · Zbl 1135.20027
[23] Deligne, P., Les immeubles des groupes de tresses généeralisées, Invent. Math., 17, 273-302, (1972) · Zbl 0238.20034
[24] Digne, F., Présentations duales des groupes de tresses de type affine Ã, Comment. Math. Helv., 81, 23-47, (2006) · Zbl 1143.20020
[25] Digne, F., A Garside presentation for Artin-Tits groups of type Cn, Ann. Inst. Fourier (Grenoble), 62, 641-666, (2012) · Zbl 1260.20056
[26] Elsner, T., Flats and the at torus theorem in systolic spaces, Geom. Topol., 13, 661-698, (2009) · Zbl 1228.20033
[27] Garside, FA, The braid group and other groups, Q. J. Math. Oxf. Ser. (2), 20, 235-254, (1969) · Zbl 0194.03303
[28] Gersten, SM; Short, H., Small cancellation theory and automatic groups. II, Invent. Math., 105, 641-662, (1991) · Zbl 0734.20014
[29] Haglund, F.: Complexes simpliciaux hyperboliques de grande dimension. Prepublication Orsay, vol. 71 (2003)
[30] Haettel, T.; Kielak, D.; Schwer, P., The 6-strand braid group is CAT(0), Geom. Dedic., 182, 263-286, (2016) · Zbl 1347.20044
[31] Hanlon, RG; Martínez-Pedroza, E., Lifting group actions, equivariant towers and subgroups of non-positively curved groups, Algebraic Geom. Topol., 14, 2783-2808, (2014) · Zbl 1335.20045
[32] Hermiller, S.; Meier, J., Algorithms and geometry for graph products of groups, J. Algebra, 171, 230-257, (1995) · Zbl 0831.20032
[33] Huang, J., Osajda, D.: Large-type Artin groups are systolic, preprint arXiv:1706.05473 (2017)
[34] Huang, J., Osajda, D.: Quasi-Euclidean tilings over 2-dimensional Artin groups and their applications, preprint arXiv:1711.00122 (2017)
[35] Januszkiewicz, T.; Świa̧tkowski, J., Simplicial nonpositive curvature, Publ. Math. Inst. Hautes Études Sci., 104, 1-85, (2006) · Zbl 1143.53039
[36] Januszkiewicz, T.; Świa̧tkowski, J., Filling invariants of systolic complexes and groups, Geom. Topol., 11, 727-758, (2007) · Zbl 1188.20043
[37] Lyndon, R.C., Schupp, P.E.: Combinatorial group theory. In: Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1977 edition · Zbl 0997.20037
[38] McCammond, J., Dual euclidean Artin groups and the failure of the lattice property, J. Algebra, 437, 308-343, (2015) · Zbl 1343.20039
[39] McCammond, J.; Sulway, R., Artin groups of Euclidean type, Invent. Math., 210, 231-282, (2017) · Zbl 1423.20032
[40] Osajda, D.; Przytycki, P., Boundaries of systolic groups, Geom. Topol., 13, 2807-2880, (2009) · Zbl 1271.20056
[41] Osajda, D.; Prytuła, T., Classifying spaces for families of sub-groups for systolic groups, Groups Geom. Dyn., 12, 1005-1060, (2018) · Zbl 06941810
[42] Peifer, D., Artin groups of extra-large type are biautomatic, J. Pure Appl. Algebra, 110, 15-56, (1996) · Zbl 0872.20036
[43] Pride, SJ, On Tits’ conjecture and other questions concerning Artin and generalized Artin groups, Invent. Math., 86, 347-356, (1986) · Zbl 0633.20021
[44] Petrunin, A., Stadler, S.: Metric minimizing surfaces revisited, preprint arXiv:1707.09635 (2017)
[45] Servatius, H., Automorphisms of graph groups, J. Algebra, 126, 34-60, (1989) · Zbl 0682.20022
[46] Świa̧tkowski, J., Regular path systems and (bi)automatic groups, Geom. Dedic., 118, 23-48, (2006) · Zbl 1165.20036
[47] VanWyk, L., Graph groups are biautomatic, J. Pure Appl. Algebra, 94, 341-352, (1994) · Zbl 0812.20018
[48] van der Lek, H.: The Homotopy Type of Complex Hyperplane Complements. Katholieke Universiteit te Nijmegen, Nijmegen (1983)
[49] Wise, D.T.: Sixtolic complexes and their fundamental groups (2003) (unpublished manuscript)
[50] Zeeman, EC, Relative simplicial approximation, Proc. Camb. Philos. Soc., 60, 39-43, (1964) · Zbl 0119.38502
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