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Algorithmic aspects of branched coverings II/V: sphere bisets and decidability of Thurston equivalence

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Abstract

We consider Thurston maps: branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is decidable. More precisely, we consider the action of mapping class groups, by pre- and post-composition, on branched coverings, and encode them algebraically as mapping class bisets. We show how the mapping class biset of maps preserving a multicurve decomposes into mapping class bisets of smaller complexity, called small mapping class bisets. We phrase the decision problem of Thurston equivalence between branched self-coverings of the sphere in terms of the conjugacy and centralizer problems in a mapping class biset. Our decomposition results on mapping class bisets reduce these decision problems to small mapping class bisets; they correspond to rational maps, homeomorphisms and maps double covered by a torus endomorphism, and their conjugacy and centralizer problems are solvable respectively in terms of complex analysis, group theory and linear algebra. Branched coverings themselves are also encoded into bisets, with actions of the fundamental groups. We characterize those bisets that arise from branched coverings between topological spheres, and extend this correspondence to maps between spheres with multicurves, whose algebraic counterparts are sphere trees of bisets. To illustrate the difference between Thurston maps and homeomorphisms, we produce a Thurston map with infinitely generated centralizer—while centralizers of homeomorphisms are always finitely generated.

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  1. Available from www-polsys.lip6.fr/~jcf/FGb/.

References

  1. Bartholdi, L., Nekrashevych, V.V.: Thurston equivalence of topological polynomials. Acta Math. 197(1), 1–51 (2006). https://doi.org/10.1007/s11511-006-0007-3. arXiv:math/0510082. https://www.gap-system.org

  2. Bartholdi, L., Dudko, D.: Algorithmic aspects of branched coverings. Ann. Fac. Sci. Toulouse 5, 1219–1296 (2017). https://doi.org/10.5802/afst.1566. arXiv:1512.05948

    Article  MathSciNet  MATH  Google Scholar 

  3. Bartholdi, L., Dudko, D.: Algorithmic aspects of branched coverings I/V Van Kampen’s theorem for bisets. Groups Geom. Dyn. 12(1), 121–172 (2018). https://doi.org/10.4171/GGD/441. arXiv:1512.08539

    Article  MathSciNet  MATH  Google Scholar 

  4. Bartholdi, L., Dudko, D.: Algorithmic aspects of branched coverings IV/V. Expanding maps. Trans. Am. Math. Soc. 370(11), 7679–7714 (2018). https://doi.org/10.1090/tran/7199. arXiv:1610.02434

    Article  MathSciNet  MATH  Google Scholar 

  5. Bartholdi, L., Mitrofanov, I.: The word and order problems for self-similar and automata groups. Groups Geom. Dyn. 14(2), 705–728. https://doi.org/10.4171/GGD/560 (2020). arXiv:1710.10109

  6. Bartholdi, L., Dudko, D.: Algorithmic aspects of branched coverings III/V. Erasing maps, orbispaces, and the Birman exact sequence (2018). arXiv:1802.03045 (submitted)

  7. Bartholdi, L., Dudko, D.: Algorithmic Aspects of Branched Coverings V/V. Symbolic and Floating-Point Algorithms (2020). (in preparation)

  8. Bartholdi, L., Buff, X.: Hurwitz Space and Its Compactification (2020). (in preparation)

  9. Bell, M.C., Webb, Richard C.H.: Polynomial-Time Algorithms for the Curve Graph (2016) arXiv:1609.09392

  10. Bestvina, M., Handel, M.: Train-tracks for surface homeomorphisms. Topology 34(1), 109–140 (1995). https://doi.org/10.1016/0040-9383(94)E0009-9

    Article  MathSciNet  MATH  Google Scholar 

  11. Birman, J.S., Series, C.: An algorithm for simple curves on surfaces. J. Lond. Math. Soc. (2) 29(2), 331–342 (1984). https://doi.org/10.1112/jlms/s2-29.2.331

    Article  MathSciNet  MATH  Google Scholar 

  12. Bonnot, S., Braverman, M., Yampolsky, M.: Thurston equivalence to a rational map is decidable. Mosc. Math. J. 12(4), 747–763, 884 (2012). https://doi.org/10.17323/1609-4514-2012-12-4-747-763 (English, with English and Russian summaries)

  13. Brown, R.: Fibrations of groupoids. J. Algebra 15, 103–132 (1970). https://doi.org/10.1016/0021-8693(70)90089-X

    Article  MathSciNet  MATH  Google Scholar 

  14. Calvez, M.: Fast Nielsen–Thurston classification of braids. Algebr. Geom. Topol. 14(3), 1745–1758 (2014). https://doi.org/10.2140/agt.2014.14.1745

    Article  MathSciNet  MATH  Google Scholar 

  15. Cannon, J.W., Floyd, W.J., Parry, W.R., Pilgrim, K.M.: Nearly Euclidean Thurston maps. Conform. Geom. Dyn. 16, 209–255 (2012). https://doi.org/10.1090/S1088-4173-2012-00248-2

    Article  MathSciNet  MATH  Google Scholar 

  16. Cohen, M., Lustig, M., Paths of geodesics and geometric intersection numbers. I, Combinatorial Group Theory and Topology (Alta, Utah, 1984), Ann. of Math. Stud., vol. 111, Princeton University Press, Princeton, NJ, pp. 479–500 (1987)

  17. Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 84, Université de Paris-Sud, Département de Mathématiques, Orsay (1984) (French)

  18. Douady, A., Hubbard, J. H.: Étude dynamique des polynômes complexes. Partie II, Publications Mathématiques d’Orsay [Mathematical Publications of Orsay], vol. 85, Université de Paris-Sud, Département de Mathématiques, Orsay (1985) (French). With the collaboration of P. Lavaurs, Tan Lei and P. Sentenac

  19. Douady, A., Hubbard, J.H.: A proof of Thurston’s topological characterization of rational functions. Acta Math. 171(2), 263–297 (1993). https://doi.org/10.1007/BF02392534

    Article  MathSciNet  MATH  Google Scholar 

  20. Epstein, D.B.A.: Curves on 2-manifolds and isotopies. Acta Math. 115, 83–107 (1966). https://doi.org/10.1007/BF02392203

    Article  MathSciNet  MATH  Google Scholar 

  21. Farb, B., Margalit, D.: A Primer on Mapping Class Groups. Princeton Mathematical Series, vol. 49. Princeton University Press, Princeton, NJ (2012)

    MATH  Google Scholar 

  22. Floyd, W., Kelsey, G., Koch, S., Lodge, R., Parry, W., Pilgrim, K.M., Saenz, E.: Origami, affine maps, and complex dynamics. Arnold Math. J. 3(3), 365–395 (2017). https://doi.org/10.1007/s40598-017-0071-0

    Article  MathSciNet  MATH  Google Scholar 

  23. Floyd, W., Parry, W., Pilgrim, K.M.: Presentations of NET maps. Fund. Math. 244(1), 49–72 (2019). https://doi.org/10.4064/fm351-11-2017. arXiv:1701.00443

    Article  MathSciNet  MATH  Google Scholar 

  24. Floyd, W., Parry, W., Pilgrim, K.M.: Modular groups, Hurwitz classes and dynamic portraits of NET maps. Groups Geom. Dyn. 13(1), 47–88 (2019). https://doi.org/10.4171/GGD/479. arXiv:1703.03983

    Article  MathSciNet  MATH  Google Scholar 

  25. de la Harpe, P.: Topics in Geometric Group Theory. University of Chicago Press, Chicago, IL (2000)

    MATH  Google Scholar 

  26. Hemion, G.: On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds. Acta Math. 142(1–2), 123–155 (1979). https://doi.org/10.1007/BF02395059

    Article  MathSciNet  MATH  Google Scholar 

  27. Hurwitz, A.: Ueber Riemann’sche Flachen mit gegebenen Verzweigungspunkten. Math. Ann. 39(1), 1–60 (1891). https://doi.org/10.1007/BF01199469. (German)

    Article  MathSciNet  MATH  Google Scholar 

  28. Kameyama, A.: The Thurston equivalence for postcritically finite branched coverings. Osaka J. Math. 38(3), 565–610 (2001). http://projecteuclid.org/euclid.ojm/1153492513

  29. Koch, S.: Teichmuller theory and critically finite endomorphisms. Adv. Math. 248, 573–617 (2013). https://doi.org/10.1016/j.aim.2013.08.019

    Article  MathSciNet  MATH  Google Scholar 

  30. Koch, S.C., Pilgrim, K.M., Selinger, N.: Fullback invariants of Thurston maps. Trans. Am. Math. Soc. 368(7), 4621–4655 (2016). https://doi.org/10.1090/tran/6482. arXiv:1212.4568

    Article  MATH  Google Scholar 

  31. Lando, S.K., Zvonkin, A.K.: Graphs on surfaces and their applications, Encyclopaedia of Mathematical Sciences, vol. 141, Springer, Berlin. With an appendix by Don B. Zagier; Low-Dimensional Topology, II (2004)

  32. Lodge, R.: Boundary values of the Thurston pullback map. Conform. Geom. Dyn. 17, 77–118 (2013). https://doi.org/10.1090/S1088-4173-2013-00255-5

    Article  MathSciNet  MATH  Google Scholar 

  33. Magnus, W., Karrass, A., Solitar, D.: Combinatorial Group Theory: Presentations of Groups in Terms of Generators and Relators. Interscience, New York (1966)

    MATH  Google Scholar 

  34. Margalit, D., McCammond, J.: Geometric presentations for the pure braid group. J. Knot Theory Ramif. 18(1), 1–20 (2009). https://doi.org/10.1142/S0218216509006859

    Article  MathSciNet  MATH  Google Scholar 

  35. McMullen, C.T.: Complex Dynamics and Renormalization. Annals of Mathematics Studies, vol. 135. Princeton University Press, Princeton, NJ (1994)

    MATH  Google Scholar 

  36. Nekrashevych, V.V.: Self-similar Groups. Mathematical Surveys and Monographs, vol. 117. American Mathematical Society, Providence, RI (2005)

    Book  Google Scholar 

  37. Nekrashevych, V.: Combinatorial models of expanding dynamical systems. Ergod. Theory Dyn. Syst. 34(3), 938–985 (2014). https://doi.org/10.1017/etds.2012.163

    Article  MathSciNet  MATH  Google Scholar 

  38. Nielsen, J.: Surface transformation classes of algebraically finite type. Danske Vid. Selsk. Math. Phys. Medd. 21(2), 89 (1944)

    MathSciNet  MATH  Google Scholar 

  39. Parry, W.R.: NETmap (2018). http://www.math.vt.edu/netmaps

  40. Pilgrim, K.M.: Combinations of Complex Dynamical Systems. Lecture Notes in Mathematics, vol. 1827. Springer, Berlin (2003)

    Book  Google Scholar 

  41. Rips, E., Sela, Z.: Cyclic splittings of finitely presented groups and the canonical JSJ decomposition. Ann. Math. (2) 146(1), 53–109 (1997). https://doi.org/10.2307/2951832

    Article  MathSciNet  MATH  Google Scholar 

  42. Selinger, N.: Thurston’s pullback map on the augmented Teichmuller space and applications. Invent. Math. 189(1), 111–142 (2012). https://doi.org/10.1007/s00222-011-0362-3

    Article  MathSciNet  MATH  Google Scholar 

  43. Selinger, N.: Topological characterization of canonical Thurston obstructions. J. Mod. Dyn. 7(1), 99–117 (2013). https://doi.org/10.3934/jmd.2013.7.99

    Article  MathSciNet  MATH  Google Scholar 

  44. Selinger, N., Yampolsky, M.: Constructive geometrization of Thurston maps and decidability of Thurston equivalence. Arnold Math. J. 1(4), 361–402 (2015). https://doi.org/10.1007/s40598-015-0024-4. arXiv:1310.1492

    Article  MathSciNet  MATH  Google Scholar 

  45. Serre, J.-P.: Trees, Springer, Berlin (1980). Translated from the French by John Stillwell

  46. The GAP Group: GAP—Groups, Algorithms, and Programming, Version 4.4.10 (2008). https://www.gap-system.org

  47. Thurston, W.P.: On the geometry and dynamics of diffeomorphisms of surfaces. Bull. Am. Math. Soc. (N.S.) 19(2), 417–431 (1988). https://doi.org/10.1090/S0273-0979-1988-15685-6

    Article  MathSciNet  MATH  Google Scholar 

  48. van Kampen, R.E.: On the connection between the fundamental groups of some related spaces. Am. J. Math. 55, 261–267 (1933). (English)

    MATH  Google Scholar 

  49. Zieschang, H., Vogt, E., Coldewey, H.-D.: Surfaces and Planar Discontinuous Groups, Lecture Notes in Mathematics, vol. 835. Springer, Berlin (1980). Translated from the German by John Stillwell

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Acknowledgements

We are grateful to Ralf Meyer for enlightening discussions on fibrations of groupoids, and to Jean–Pierre Spaenlehauer for help in computing the equation of the algebraic correspondence in Sect. 9.2.2.

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Authors and Affiliations

  1. Mathematisches Institut, Georg-August Universität zu Göttingen, Göttingen, Germany

    Laurent Bartholdi

  2. École Normale Supérieure, Paris, France

    Laurent Bartholdi

  3. Stony Brook University, Stony Brook, USA

    Dzmitry Dudko

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Bartholdi, L., Dudko, D. Algorithmic aspects of branched coverings II/V: sphere bisets and decidability of Thurston equivalence. Invent. math. 223, 895–994 (2021). https://doi.org/10.1007/s00222-020-00995-2

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