Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ON TRANSLATION LENGTHS OF PSEUDO-ANOSOV MAPS ON THE CURVE GRAPH

  • Received : 2023.02.14
  • Accepted : 2024.01.12
  • Published : 2024.05.31
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Abstract

We show that a pseudo-Anosov map constructed as a product of the large power of Dehn twists of two filling curves always has a geodesic axis on the curve graph of the surface. We also obtain estimates of the stable translation length of a pseudo-Anosov map, when two filling curves are replaced by multicurves. Three main applications of our theorem are the following: (a) determining which word realizes the minimal translation length on the curve graph within a specific class of words, (b) giving a new class of pseudo-Anosov maps optimizing the ratio of stable translation lengths on the curve graph to that on Teichmüller space, (c) giving a partial answer of how much power is needed for Dehn twists to generate right-angled Artin subgroup of the mapping class group.

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References

  1. T. Aougab and S. J. Taylor, Small intersection numbers in the curve graph, Bull. Lond. Math. Soc. 46 (2014), no. 5, 989-1002. https://doi.org/10.1112/blms/bdu057 
  2. T. Aougab and S. J. Taylor, Pseudo-Anosovs optimizing the ratio of Teichmuller to curve graph translation length, in In the tradition of Ahlfors-Bers. VII, 17-28, Contemp. Math., 696, Amer. Math. Soc., Providence, RI, 2017. https://doi.org/10.1090/conm/696/14014 
  3. H. Baik, C. Kim, S. Kwak, and H. Shin, On translation lengths of Anosov maps on the curve graph of the torus, Geom. Dedicata 214 (2021), 399-426. https://doi.org/10.1007/s10711-021-00622-1 
  4. B. H. Bowditch, Tight geodesics in the curve complex, Invent. Math. 171 (2008), no. 2, 281-300. https://doi.org/10.1007/s00222-007-0081-y 
  5. B. Farb and D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton Univ. Press, Princeton, NJ, 2012. 
  6. V. Gadre, E. Hironaka, R. P. Kent IV, and C. J. Leininger, Lipschitz constants to curve complexes, Math. Res. Lett. 20 (2013), no. 4, 647-656. https://doi.org/10.4310/MRL.2013.v20.n4.a4 
  7. T. Koberda, Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups, Geom. Funct. Anal. 22 (2012), no. 6, 1541-1590. https://doi.org/10.1007/s00039-012-0198-z 
  8. J. Mangahas, A recipe for short-word pseudo-Anosovs, Amer. J. Math. 135 (2013), no. 4, 1087-1116. https://doi.org/10.1353/ajm.2013.0037 
  9. H. A. Masur and Y. N. Minsky, Geometry of the complex of curves. II. Hierarchical structure, Geom. Funct. Anal. 10 (2000), no. 4, 902-974. https://doi.org/10.1007/PL00001643 
  10. R. C. Penner, A construction of pseudo-Anosov homeomorphisms, Trans. Amer. Math. Soc. 310 (1988), no. 1, 179-197. https://doi.org/10.2307/2001116 
  11. I. Runnels, Effective generation of right-angled Artin groups in mapping class groups, Geom. Dedicata 214 (2021), 277-294. https://doi.org/10.1007/s10711-021-00615-0 
  12. D. Seo, Powers of Dehn twists generating right-angled Artin groups, Algebraic & Geometric Topology 21 (2021), no. 3, 1511-1533.  https://doi.org/10.2140/agt.2021.21.1511
  13. W. P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417-431. https://doi.org/10.1090/S0273-0979-1988-15685-6 
  14. Y. Watanabe, Pseudo-Anosov mapping classes from pure mapping classes, Trans. Amer. Math. Soc. 373 (2020), no. 1, 419-434. https://doi.org/10.1090/tran/7919 
  15. R. C. H. Webb, Uniform bounds for bounded geodesic image theorems, J. Reine Angew. Math. 709 (2015), 219-228. https://doi.org/10.1515/crelle-2013-0109