Abstract
Given two conjugate mapping classes f and g, we produce a conjugating element ω such that \({|\omega| \le K (|f| + |g|)}\), where | · | denotes the word metric with respect to a fixed generating set, and K is a constant depending only on the generating set. As a consequence, the conjugacy problem for mapping class groups is exponentially bounded.
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References
I. Agol. Ideal Triangulations of Pseudo-Anosov Mapping Tori, Topology and Geometry in Dimension Three. Contemp. Math., Vol. 560. Am. Math. Soc., Providence, RI (2011), pp. 1–17.
Bestvina M., Handel M (1995) Train-tracks for surface homeomorphisms. Topology, 1(34): 109–140
M.R. Bridson and A. Haefliger. Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 319. Springer, Berlin (1999).
Behrstock J., Kleiner B., Minsky Y., Mosher L (2012) Geometry and rigidity of mapping class groups. Geometric Topology, 2(16): 781–888
Brock J., Masur H (2008) Coarse and synthetic Weil–Petersson geometry: Quasi-flats, geodesics and relative hyperbolicity. Geometric Topology, 4(12): 2453–2495
Boone W.W (1959) The word problem. Annals of Mathematics, 2(70): 207–265
Bowditch B.H (2006) Intersection numbers and the hyperbolicity of the curve complex. Journal für die Reine und Angewandte Mathematik, 598: 105–129
M. Dehn (1911) Über unendliche diskontinuierliche Gruppen. Mathematische Annalen, 1(71): 116–144
D.B.A. Epstein, J.W. Cannon, D.F. Holt, S.V.F. Levy, M.S. Paterson, and W.P. Thurston. Word Processing in Groups. Jones and Bartlett Publishers, Boston (1992).
B. Farb. Some Problems on Mapping Class Groups and Moduli Space, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., Vol. 74. Am. Math. Soc., Providence, RI (2006), pp. 11–55.
Travaux de Thurston sur les surfaces, Astérisque, Vol. 66, Société Mathématique de France, Paris (1979), Séminaire Orsay, With an English summary.
B. Farb and D. Margalit. A primer on Mapping Class Groups, Princeton Mathematical Series, Vol. 49. Princeton University Press, Princeton (2012).
M. Gromov. Hyperbolic Groups, Essays in group theory, Mathematical Sciences Research Institute Publications, Vol. 8. Springer, New York (1987), pp. 75–263.
E.K. Grossman. On the residual finiteness of certain mapping class groups. Journal of the London Mathematical Society, (2)9 (1974/1975), 160–164.
U. Hamenstädt. Geometry of the mapping class group, II: A biautomatic structure, Preprint, 2009, Available at arXiv:math/0912.0137v1 [math.GR].
W.J. Harvey. Boundary Structure of the Modular Group, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), Annals of Math. Stud., Vol. 97. Princeton Univ. Press, Princeton, N.J. (1981), pp. 245–251.
Hemion G (1979) On the classification of homeomorphisms of 2-manifolds and the classification of 3-manifolds.. Acta Mathematica, (1−2)142: 123–155
Hatcher A., Thurston W (1980) A presentation for the mapping class group of a closed orientable surface. Topology, 3(19): 221–237
Kerckhoff S.P (1983) The Nielsen realization problem. The Annals of Mathematics, 2(117): 235–265
Lickorish W.B.R (1964) A finite set of generators for the homeotopy group of a 2-manifold. Proceedings of the Cambridge Philosophical Society Journal, 60: 769–778
I.G. Lysënok. Some algorithmic properties of hyperbolic groups. Izvestiya Akademii Nauk SSSR Seriya Matematicheskaya,4)53 (1989), 814–832, 912.
J.F. Manning. Geometry of pseudocharacters. Geometric Topology , 9 (2005), 1147–1185 (electronic).
Y.N. Minsky. A Geometric Approach to the Complex of Curves on a Surface, Topology and Teichmüller spaces (Katinkulta, 1995). World Sci. Publ., River Edge, NJ (1996), pp. 149–158.
Y. Minsky. The classification of Kleinian surface groups. I. Models and bounds. Annals of Mathematics (2), (1)171 (2010), 1–107.
Masur H.A., Minsky Y.N (1999) Geometry of the complex of curves. I. Hyperbolicity. Inventiones Mathematicae, 1(138): 103–149
Masur H.A., Minsky Y.N (2000) Geometry of the complex of curves. II. Hierarchical structure. Geomertic and Functional Analysis, 4(10): 902–974
L. Mosher. The Classification of Pseudo-Anosovs, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., Vol. 112. Cambridge University Press, Cambridge (1986), pp. 13–75.
L. Mosher. Mapping class groups are automatic. Annals of Mathematics (2), (2)142 (1995), 303–384.
L. Mosher. Train track expansions of measured foliations, Preprint, 2003, Available at andromeda.rutgers.edu/~mosher.
L. Mosher. MSRI Course on mapping class groups, Lecture notes, 2007, Available at andromeda.rutger.edu/~mosher.
P.S. Novikov. On the Algorithmic Insolvability of the Word Problem in Group Theory, American Mathematical Society Translations, Ser 2, Vol. 9. American Mathematical Society, Providence, RI (1958), pp. 1–122.
K. Rafi and S. Schleimer. Covers and the curve complex. Geometric Topology, (4)13 (2009), 2141–2162.
W.P. Thurston. On the geometry and dynamics of diffeomorphisms of surfaces. Bulletin of the American Mathematical Society (N.S.), (2)19 (1988), 417–431.
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Tao, J. Linearly Bounded Conjugator Property for Mapping Class Groups. Geom. Funct. Anal. 23, 415–466 (2013). https://doi.org/10.1007/s00039-012-0206-3
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DOI: https://doi.org/10.1007/s00039-012-0206-3