Lower bounds on cubical dimensionof $C’(1/6)$ groups
HTML articles powered by AMS MathViewer
- by Kasia Jankiewicz PDF
- Proc. Amer. Math. Soc. 148 (2020), 3293-3306 Request permission
Abstract:
For each $n$ we construct examples of finitely presented $C’(1/6)$ small cancellation groups that do not act properly on any $n$-dimensionalCAT(0) cube complex.References
- Noel Brady and John Crisp, Two-dimensional Artin groups with $\textrm {CAT}(0)$ dimension three, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), 2002, pp. 185–214. MR 1950878, DOI 10.1023/A:1020962804856
- Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
- Martin R. Bridson, Length functions, curvature and the dimension of discrete groups, Math. Res. Lett. 8 (2001), no. 4, 557–567. MR 1851271, DOI 10.4310/MRL.2001.v8.n4.a14
- Samuel Brown, $\mathrm {CAT}(-1)$ metrics on small cancellation groups, arXiv:1607.02580, pages 1–12, 2016.
- Victor Chepoi and Mark F. Hagen, On embeddings of CAT(0) cube complexes into products of trees via colouring their hyperplanes, J. Combin. Theory Ser. B 103 (2013), no. 4, 428–467. MR 3071375, DOI 10.1016/j.jctb.2013.04.003
- John Crisp, On the $\textrm {CAT}(0)$ dimension of 2-dimensional Bestvina-Brady groups, Algebr. Geom. Topol. 2 (2002), 921–936. MR 1936975, DOI 10.2140/agt.2002.2.921
- Pierre-Emmanuel Caprace and Michah Sageev, Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21 (2011), no. 4, 851–891. MR 2827012, DOI 10.1007/s00039-011-0126-7
- Talia Fernós, Max Forester, and Jing Tao, Effective quasimorphisms on right-angled Artin groups, Ann. Inst. Fourier (Grenoble) 69 (2019), no. 4, 1575–1626 (English, with English and French summaries). MR 4010865
- Anthony Genevois, Rank-one isometries of $\mathrm {CAT(0)}$ cube complexes and their centralisers, arXiv:1905.00735, pages 1–20, 2019.
- M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR 919829, DOI 10.1007/978-1-4613-9586-7_{3}
- M. Gromov, $\textrm {CAT}(\kappa )$-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), no. Geom. i Topol. 7, 100–140, 299–300 (English, with Russian summary); English transl., J. Math. Sci. (N.Y.) 119 (2004), no. 2, 178–200. MR 1879258, DOI 10.1023/B:JOTH.0000008756.15786.0f
- Ronald L. Graham, Bruce L. Rothschild, and Joel H. Spencer, Ramsey theory, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Inc., New York, 1980. MR 591457
- Thomas Haettel, Virtually cocompactly cubulated Artin-Tits groups, arXiv:1509.08711, pages 1–25, 2017.
- Frédéric Haglund, Isometries of CAT(0) cube complexes are semisimple, pages 1–17, 2007. Preprint.
- Jingyin Huang, Kasia Jankiewicz, and Piotr Przytycki, Cocompactly cubulated 2-dimensional Artin groups, Comment. Math. Helv. 91 (2016), no. 3, 519–542. MR 3541719, DOI 10.4171/CMH/394
- Kasia Jankiewicz and Daniel T. Wise, Cubulating small cancellation free products, pages 1–11, 2017. Preprint.
- Marcin Kotowski and MichałKotowski, Random groups and property $(T)$: Żuk’s theorem revisited, J. Lond. Math. Soc. (2) 88 (2013), no. 2, 396–416. MR 3106728, DOI 10.1112/jlms/jdt024
- Aditi Kar and Michah Sageev, Uniform exponential growth for CAT(0) square complexes, Algebr. Geom. Topol. 19 (2019), no. 3, 1229–1245. MR 3954280, DOI 10.2140/agt.2019.19.1229
- Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89, Springer-Verlag, Berlin-New York, 1977. MR 0577064
- Alexandre Martin, Complexes of groups and geometric small cancelation over graphs of groups, Bull. Soc. Math. France 145 (2017), no. 2, 193–223 (English, with English and French summaries). MR 3749783, DOI 10.24033/bsmf.2734
- Stephen J. Pride, Some finitely presented groups of cohomological dimension two with property (FA), J. Pure Appl. Algebra 29 (1983), no. 2, 167–168. MR 707619, DOI 10.1016/0022-4049(83)90105-6
- Michah Sageev, Ends of group pairs and non-positively curved cube complexes, Proc. London Math. Soc. (3) 71 (1995), no. 3, 585–617. MR 1347406, DOI 10.1112/plms/s3-71.3.585
- Michah Sageev, $\rm CAT(0)$ cube complexes and groups, Geometric group theory, IAS/Park City Math. Ser., vol. 21, Amer. Math. Soc., Providence, RI, 2014, pp. 7–54. MR 3329724, DOI 10.1090/pcms/021/02
- D. T. Wise, Cubulating small cancellation groups, Geom. Funct. Anal. 14 (2004), no. 1, 150–214. MR 2053602, DOI 10.1007/s00039-004-0454-y
- Daniel J. Woodhouse, A generalized axis theorem for cube complexes, Algebr. Geom. Topol. 17 (2017), no. 5, 2737–2751. MR 3704240, DOI 10.2140/agt.2017.17.2737
- A. Żuk, Property (T) and Kazhdan constants for discrete groups, Geom. Funct. Anal. 13 (2003), no. 3, 643–670. MR 1995802, DOI 10.1007/s00039-003-0425-8
Additional Information
- Kasia Jankiewicz
- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois 60637
- MR Author ID: 1148637
- Email: kasia@math.uchicago.edu
- Received by editor(s): July 26, 2019
- Received by editor(s) in revised form: November 13, 2019, and December 24, 2019
- Published electronically: May 8, 2020
- Additional Notes: The author was partially supported by (Polish) Narodowe Centrum Nauki, grant no. UMO-2015/18/M/ST1/00050.
- Communicated by: David Futer
- © Copyright 2020 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 148 (2020), 3293-3306
- MSC (2010): Primary 20F65, 20F67
- DOI: https://doi.org/10.1090/proc/15013
- MathSciNet review: 4108839