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Lower bounds on cubical dimensionof \(C'(1/6)\) groups. (English) Zbl 07211060
Summary: For each \(n\) we construct examples of finitely presented \(C'(1/6)\) small cancellation groups that do not act properly on any \(n\)-dimensional CAT(0) cube complex.
MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
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[1] Brady, Noel; Crisp, John, Two-dimensional Artin groups with \(\operatorname{CAT}(0)\) dimension three, Geom. Dedicata. Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 94, 185-214 (2002) · Zbl 1070.20043
[2] Bridson, Martin R.; Haefliger, Andr\'e, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319, xxii+643 pp. (1999), Springer-Verlag, Berlin · Zbl 0988.53001
[3] Bridson, Martin R., Length functions, curvature and the dimension of discrete groups, Math. Res. Lett., 8, 4, 557-567 (2001) · Zbl 0990.20026
[4] [Bro16]Brown16 Samuel Brown, \emph \(\text CAT(-1)\) metrics on small cancellation groups, \arXiv 1607.02580, pages 1-12, 2016.
[5] Chepoi, Victor; Hagen, Mark F., On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes, J. Combin. Theory Ser. B, 103, 4, 428-467 (2013) · Zbl 1301.05120
[6] Crisp, John, On the \(\operatorname{CAT}(0)\) dimension of 2-dimensional Bestvina-Brady groups, Algebr. Geom. Topol., 2, 921-936 (2002) · Zbl 1055.20036
[7] Caprace, Pierre-Emmanuel; Sageev, Michah, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal., 21, 4, 851-891 (2011) · Zbl 1266.20054
[8] Fern\'os, Talia; Forester, Max; Tao, Jing, Effective quasimorphisms on right-angled Artin groups, Ann. Inst. Fourier (Grenoble), 69, 4, 1575-1626 (2019) · Zbl 07114096
[9] [Gen19]Genevois19 Anthony Genevois, \emph Rank-one isometries of \(\text CAT(0)\) cube complexes and their centralisers, \arXiv 1905.00735, pages 1-20, 2019.
[10] Gromov, M., Hyperbolic groups. Essays in group theory, Math. Sci. Res. Inst. Publ. 8, 75-263 (1987), Springer, New York
[11] Gromov, M., \( \operatorname{CAT}(\kappa)\)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI). J. Math. Sci. (N.Y.), 280 119, 2, 178-200 (2004) · Zbl 1089.53029
[12] \bibGrahamRothschildSpencer80book label=GRS80, author=Graham, Ronald L., author=Rothschild, Bruce L., author=Spencer, Joel H., title=Ramsey theory, pages=ix+174, publisher=John Wiley & Sons, Inc., New York, date=1980, note=Wiley-Interscience Series in Discrete Mathematics; A Wiley-Interscience Publication, isbn=0-471-05997-8, review=\MR591457,
[13] [Hae17]HaettelArtin Thomas Haettel, \emph Virtually cocompactly cubulated Artin-Tits groups, \arXiv 1509.08711, pages 1-25, 2017.
[14] [Hag07]HaglundSemiSimple Fr\'ed\'eric Haglund, \emph Isometries of CAT(0) cube complexes are semisimple, pages 1-17, 2007. Preprint.
[15] Huang, Jingyin; Jankiewicz, Kasia; Przytycki, Piotr, Cocompactly cubulated 2-dimensional Artin groups, Comment. Math. Helv., 91, 3, 519-542 (2016) · Zbl 1401.20044
[16] [JW17]JankiewiczWise17 Kasia Jankiewicz and Daniel T. Wise, \emph Cubulating small cancellation free products, pages 1-11, 2017. Preprint.
[17] Kotowski, Marcin; Kotowski, Micha\l, Random groups and property \((T)\): \.Zuk’s theorem revisited, J. Lond. Math. Soc. (2), 88, 2, 396-416 (2013) · Zbl 1309.20035
[18] Kar, Aditi; Sageev, Michah, Uniform exponential growth for CAT(0) square complexes, Algebr. Geom. Topol., 19, 3, 1229-1245 (2019) · Zbl 07142601
[19] Lyndon, Roger C.; Schupp, Paul E., Combinatorial group theory, xiv+339 pp. (1977), Springer-Verlag, Berlin-New York · Zbl 0997.20037
[20] Martin, Alexandre, Complexes of groups and geometric small cancelation over graphs of groups, Bull. Soc. Math. France, 145, 2, 193-223 (2017) · Zbl 1446.20066
[21] Pride, Stephen J., Some finitely presented groups of cohomological dimension two with property (FA), J. Pure Appl. Algebra, 29, 2, 167-168 (1983) · Zbl 0513.20019
[22] Sageev, Michah, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3), 71, 3, 585-617 (1995) · Zbl 0861.20041
[23] Sageev, Michah, \( \rm CAT(0)\) cube complexes and groups. Geometric group theory, IAS/Park City Math. Ser. 21, 7-54 (2014), Amer. Math. Soc., Providence, RI
[24] Wise, D. T., Cubulating small cancellation groups, Geom. Funct. Anal., 14, 1, 150-214 (2004) · Zbl 1071.20038
[25] Woodhouse, Daniel J., A generalized axis theorem for cube complexes, Algebr. Geom. Topol., 17, 5, 2737-2751 (2017) · Zbl 1405.20030
[26] \.Zuk, A., Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal., 13, 3, 643-670 (2003) · Zbl 1036.22004
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