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The seven-strand braid group is CAT(0)

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Abstract

We prove that the 7-strand braid group is CAT(0) by elaborating on the argument of Haettel, Kielak, and Schwer.

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References

  1. Charney, R.: An introduction to right-angled Artin groups. Geom. Dedicata. 125, 141–158 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Brady, T., McCammond, J.: Braids, posets and orthoschemes. Algebr. Geom. Topol. 10(4), 2277–2314 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  3. Haettel, T., Kielak, D., Schwer, P.: The 6-strand braid group is CAT(0). Geom Dedicata. 182, 263–286 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  4. Gromov, M.: Hyperbolic groups. In: Essays in group theory, volume 8 of Mathematical Sciences Research Institute Publications, pp. 75-263. Springer, New York, (1987)

  5. Bowditch, B. H.: Notes on locally CAT(1) spaces. In: Geometric group theory (Columbus, OH, 1992), volume 3 of Ohio State Univ. Math. Res. Inst. Publ., pp. 1–48. de Gruyter, Berlin (1995)

  6. Charney, R., Davis, M.: Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds. Am. J. Math. 115(5), 929–1009 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  7. Haettel, T.: Cocompactly cubulated Artin-Tits groups, ArXiv e-prints (2015)

  8. Birman, J.S., Ko, K.H., Lee, S.J.: A new approach to the word and conjugacy problem in the braid groups. Adv. Math. 139, 322–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  9. Elrifai, E.A., Morton, H.R.: Algorithms for positive braids. Quart. J. Math. Oxford 45, 479–497 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  10. Brady, T.: A partial order on the symmetric group and new K($\pi $, 1)’s for the braid groups. Adv. Math. 161(1), 20–40 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature. Grundlegren der Mathematischen Wissenschaften, vol. 319. Springer Verlag, Berlin (1999)

    Book  MATH  Google Scholar 

  12. Abramenko, P., Brown, K.S.: Buildings. Graduate Texts in Mathematics. Theory and Applications, vol. 248. Springer, New York (2008)

    MATH  Google Scholar 

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Acknowledgements

I would like to thank Professor Ki Hyung Ko. Whenever research progress was not going well, he suggested a way to think in different directions. So I was able to get research results. In addition, I would like to thank Professor Hyung Ryeol Baik for helping me a lot with the completion and publication process of this thesis. I would like to express my gratitude to my grandmother, parents, and the older brother who always supported and waited for me, and to the members of the lab who always treated me well and gave me strength.

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  1. Department of Mathematics, Korea Advanced Institute of Science and Technology, Daejeon, 307-701, Korea

    Seong Gu Jeong

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  1. Seong Gu Jeong
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Correspondence to Seong Gu Jeong.

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Jeong, S.G. The seven-strand braid group is CAT(0). manuscripta math. 171, 563–581 (2023). https://doi.org/10.1007/s00229-022-01387-x

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  • DOI: https://doi.org/10.1007/s00229-022-01387-x

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