Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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UNIQUENESS OF QUASI-ROOTS IN RIGHT-ANGLED ARTIN GROUPS

  • Lee, Eon-Kyung (Department of Mathematics and Statistics Sejong University) ;
  • Lee, Sang-Jin (Department of Mathematics Konkuk University)
  • Received : 2021.09.16
  • Accepted : 2022.05.13
  • Published : 2022.07.01
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Abstract

We introduce the notion of quasi-roots and study their uniqueness in right-angled Artin groups.

Keywords

Acknowledgement

This work was supported by the National Research Foundation of Korea (NRF-2018R1D1A1B07043291 and NRF-2018R1D1A1B07043268).

References

  1. G. Baumslag, Some aspects of groups with unique roots, Acta Math. 104 (1960), 217-303. https://doi.org/10.1007/BF02546390
  2. C. Bonatti and L. Paris, Roots in the mapping class groups, Proc. Lond. Math. Soc. (3) 98 (2009), no. 2, 471-503. https://doi.org/10.1112/plms/pdn036
  3. M. R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften, 319, Springer-Verlag, Berlin, 1999. https://doi.org/10.1007/978-3-662-12494-9
  4. R. Charney, An introduction to right-angled Artin groups, Geom. Dedicata 125 (2007), 141-158. https://doi.org/10.1007/s10711-007-9148-6
  5. J. Crisp, E. Godelle, and B. Wiest, The conjugacy problem in subgroups of right-angled Artin groups, J. Topol. 2 (2009), no. 3, 442-460. https://doi.org/10.1112/jtopol/jtp018
  6. G. Duchamp and J.-Y. Thibon, Simple orderings for free partially commutative groups, Internat. J. Algebra Comput. 2 (1992), no. 3, 351-355. https://doi.org/10.1142/S0218196792000219
  7. E. Ghys and P. de la Harpe, Espaces metriques hyperboliques, in Sur les groupes hyperboliques d'apres Mikhael Gromov (Bern, 1988), 27-45, Progr. Math., 83, Birkhauser Boston, Boston, MA, 1990. https://doi.org/10.1007/978-1-4684-9167-8_2
  8. J. Gonz'alez-Meneses, The nth root of a braid is unique up to conjugacy, Algebr. Geom. Topol. 3 (2003), 1103-1118. https://doi.org/10.2140/agt.2003.3.1103
  9. S. Kim and T. Koberda, Embedability between right-angled Artin groups, Geom. Topol. 17 (2013), no. 1, 493-530. https://doi.org/10.2140/gt.2013.17.493
  10. S. Kim and T. Koberda, The geometry of the curve graph of a right-angled Artin group, Internat. J. Algebra Comput. 24 (2014), no. 2, 121-169. https://doi.org/10.1142/S021819671450009X
  11. S. Kim and T. Koberda, Anti-trees and right-angled Artin subgroups of braid groups, Geom. Topol. 19 (2015), no. 6, 3289-3306. https://doi.org/10.2140/gt.2015.19.3289
  12. D. M. Kim and D. Rolfsen, An ordering for groups of pure braids and fibre-type hyperplane arrangements, Canad. J. Math. 55 (2003), no. 4, 822-838. https://doi.org/10.4153/CJM-2003-034-2
  13. E.-K. Lee and S.-J. Lee, Uniqueness of roots up to conjugacy for some affine and finite type Artin groups, Math. Z. 265 (2010), no. 3, 571-587. https://doi.org/10.1007/s00209-009-0530-y
  14. E.-K. Lee and S.-J. Lee, Path lifting properties and embedding between RAAGs, J. Algebra 448 (2016), 575-594. https://doi.org/10.1016/j.jalgebra.2015.09.010
  15. E.-K. Lee and S.-J. Lee, Embeddability of right-angled Artin groups on complements of trees, Internat. J. Algebra Comput. 28 (2018), no. 3, 381-394. https://doi.org/10.1142/S0218196718500182
  16. G. S. Makanin, The normalizers of a braid group, Mat. Sb. (N.S.) 86(128) (1971), 171-179.
  17. A. I. Mal'cev, Nilpotent torsion-free groups, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949), 201-212.
  18. D. Rolfsen and J. Zhu, Braids, orderings and zero divisors, J. Knot Theory Ramifications 7 (1998), no. 6, 837-841. https://doi.org/10.1142/S0218216598000425
  19. H. Servatius, Automorphisms of graph groups, J. Algebra 126 (1989), no. 1, 34-60. https://doi.org/10.1016/0021-8693(89)90319-0