Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Pullback invariants of Thurston maps


Authors: Sarah Koch, Kevin M. Pilgrim and Nikita Selinger
Journal: Trans. Amer. Math. Soc. 368 (2016), 4621-4655
MSC (2010): Primary 30F60, 32G15, 37F20
DOI: https://doi.org/10.1090/tran/6482
Published electronically: September 24, 2015
MathSciNet review: 3456156
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Associated to a Thurston map $ f: S^2 \to S^2$ with postcritical set $ P$ are several different invariants obtained via pullback: a relation $ \mathcal {S}_{P} {\stackrel {f}{\longleftarrow }} \mathcal {S}_{P}$ on the set $ \mathcal {S}_{P}$ of free homotopy classes of curves in $ S^2\setminus P$, a linear operator $ \lambda _f: \mathbb{R}[\mathcal {S}_{P}]\to \mathbb{R}[\mathcal {S}_{P}]$ on the free $ \mathbb{R}$-module generated by $ \mathcal {S}_{P}$, a virtual endomorphism $ \phi _f: \mathrm {PMod}(S^2, P) \dashrightarrow \mathrm {PMod}(S^2, P)$ on the pure mapping class group, an analytic self-map $ \sigma _f: \mathcal {T}(S^2, P) \to \mathcal {T}(S^2, P)$ of an associated Teichmüller space, and an analytic self-correspondence $ X\circ Y^{-1}: \mathcal {M}(S^2, P) \rightrightarrows \mathcal {M}(S^2, P)$ of an associated moduli space. Viewing these associated maps as invariants of $ f$, we investigate relationships between their properties.


References [Enhancements On Off] (What's this?)

  • [BN1] Laurent Bartholdi and Volodymyr Nekrashevych, Thurston equivalence of topological polynomials, Acta Math. 197 (2006), no. 1, 1-51. MR 2285317 (2008c:37072), https://doi.org/10.1007/s11511-006-0007-3
  • [BN2] Laurent Bartholdi and Volodymyr V. Nekrashevych, Iterated monodromy groups of quadratic polynomials. I, Groups Geom. Dyn. 2 (2008), no. 3, 309-336. MR 2415302 (2009d:37074), https://doi.org/10.4171/GGD/42
  • [BM] Robert W. Bell and Dan Margalit, Injections of Artin groups, Comment. Math. Helv. 82 (2007), no. 4, 725-751. MR 2341838 (2009b:20062), https://doi.org/10.4171/CMH/108
  • [BLM] Joan S. Birman, Alex Lubotzky, and John McCarthy, Abelian and solvable subgroups of the mapping class groups, Duke Math. J. 50 (1983), no. 4, 1107-1120. MR 726319 (85k:20126), https://doi.org/10.1215/S0012-7094-83-05046-9
  • [BEKP] Xavier Buff, Adam Epstein, Sarah Koch, and Kevin Pilgrim, On Thurston's pullback map, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 561-583. MR 2508269 (2010g:37071), https://doi.org/10.1201/b10617-20
  • [CFP] J. W. Cannon, W. J. Floyd, and W. R. Parry, Finite subdivision rules, Conform. Geom. Dyn. 5 (2001), 153-196 (electronic). MR 1875951 (2002j:52021), https://doi.org/10.1090/S1088-4173-01-00055-8
  • [CFPP] J. W. Cannon, W. J. Floyd, W. R. Parry, and K. M. Pilgrim, Subdivision rules and virtual endomorphisms, Geom. Dedicata 141 (2009), 181-195. MR 2520071 (2010d:37086), https://doi.org/10.1007/s10711-009-9352-7
  • [DH] Adrien Douady and John H. Hubbard, A proof of Thurston's topological characterization of rational functions, Acta Math. 171 (1993), no. 2, 263-297. MR 1251582 (94j:58143), https://doi.org/10.1007/BF02392534
  • [FLP] Albert Fathi, François Laudenbach, and Valentin Poénaru, Thurston's work on surfaces, Mathematical Notes, vol. 48, Princeton University Press, Princeton, NJ, 2012. Translated from the 1979 French original by Djun M. Kim and Dan Margalit. MR 3053012
  • [HP1] Peter Haïssinsky and Kevin M. Pilgrim, Finite type coarse expanding conformal dynamics, Groups Geom. Dyn. 5 (2011), no. 3, 603-661. MR 2813529, https://doi.org/10.4171/GGD/141
  • [HP2] Peter Haïssinsky and Kevin M. Pilgrim, An algebraic characterization of expanding Thurston maps, J. Mod. Dyn. 6 (2012), no. 4, 451-476. MR 3008406
  • [HK] John H. Hubbard and Sarah Koch, An analytic construction of the Deligne-Mumford compactification of the moduli space of curves, J. Differential Geom. 98 (2014), no. 2, 261-313. MR 3263519
  • [Hub] John Hamal Hubbard, Teichmüller theory and applications to geometry, topology, and dynamics. Vol. 1,. Teichmüller theory. With contributions by Adrien Douady, William Dunbar, Roland Roeder, Sylvain Bonnot, David Brown, Allen Hatcher, Chris Hruska and Sudeb Mitra. With forewords by William Thurston and Clifford Earle. Matrix Editions, Ithaca, NY, 2006. MR 2245223 (2008k:30055)
  • [Iva1] Nikolai V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs, vol. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. MR 1195787 (93k:57031)
  • [Iva2] Nikolai V. Ivanov,
    Mapping class groups,
    http://www.math.msu.edu/~ivanov/m99.ps, 1998.
  • [Kam] Atsushi Kameyama, The Thurston equivalence for postcritically finite branched coverings, Osaka J. Math. 38 (2001), no. 3, 565-610. MR 1860841 (2002h:57004)
  • [Kel] Gregory A. Kelsey, Mapping schemes realizable by obstructed topological polynomials, Conform. Geom. Dyn. 16 (2012), 44-80. MR 2893472, https://doi.org/10.1090/S1088-4173-2012-00239-1
  • [Koc1] Sarah Koch, Teichmüller theory and endomorphisms of $ \mathbb{P}^n$, PhD thesis, University of Provence, 2007.
  • [Koc2] Sarah Koch, Teichmüller theory and critically finite endomorphisms, Adv. Math. 248 (2013), 573-617. MR 3107522, https://doi.org/10.1016/j.aim.2013.08.019
  • [KPS2] Sarah Koch, Kevin M. Pilgrim, and Nikita Selinger,
    Limit sets of Thurston pullback semigroups,
    Manuscript, 2014.
  • [Lod] Russell Lodge, Boundary values of the Thurston pullback map, ProQuest LLC, Ann Arbor, MI, 2012. Thesis (Ph.D.)-Indiana University. MR 3054977
  • [Mas] Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623-635. MR 0417456 (54 #5506)
  • [Nek1] Volodymyr Nekrashevych, Combinatorics of polynomial iterations, Complex dynamics, A K Peters, Wellesley, MA, 2009, pp. 169-214. MR 2508257 (2011k:37082), https://doi.org/10.1201/b10617-5
  • [Nek2] Volodymyr Nekrashevych, Combinatorial models of expanding dynamical systems, Ergodic Theory Dynam. Systems 34 (2014), no. 3, 938-985. MR 3199801, https://doi.org/10.1017/etds.2012.163
  • [Pil1] Kevin M. Pilgrim, Canonical Thurston obstructions, Adv. Math. 158 (2001), no. 2, 154-168. MR 1822682 (2001m:57004), https://doi.org/10.1006/aima.2000.1971
  • [Pil2] Kevin M. Pilgrim, Combination, decomposition, and structure theory for postcritically finite branched coverings of the two-sphere to itself,
    to appear, Springer Lecture Notes in Math., 2003.
  • [Pil3] Kevin M. Pilgrim, An algebraic formulation of Thurston's characterization of rational functions, Ann. Fac. Sci. Toulouse Math. (6) 21 (2012), no. 5, 1033-1068 (English, with English and French summaries). MR 3088266, https://doi.org/10.5802/afst.1361
  • [Sel1] Nikita Selinger, Thurston's pullback map on the augmented Teichmüller space and applications, Invent. Math. 189 (2012), no. 1, 111-142. MR 2929084, https://doi.org/10.1007/s00222-011-0362-3
  • [Sel2] Nikita Selinger, Topological characterization of canonical Thurston obstructions, J. Mod. Dyn. 7 (2013), no. 1, 99-117. MR 3071467, https://doi.org/10.3934/jmd.2013.7.99
  • [Wol] Scott A. Wolpert, The Weil-Petersson metric geometry, Handbook of Teichmüller theory. Vol. II, IRMA Lect. Math. Theor. Phys., vol. 13, Eur. Math. Soc., Zürich, 2009, pp. 47-64. MR 2497791 (2010i:32012), https://doi.org/10.4171/055-1/2

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 30F60, 32G15, 37F20

Retrieve articles in all journals with MSC (2010): 30F60, 32G15, 37F20


Additional Information

Sarah Koch
Affiliation: Department of Mathematics, University of Michigan, 530 Church Street, Ann Arbor, Michigan 48109

Kevin M. Pilgrim
Affiliation: Department of Mathematics, Indiana University, 831 E. Third Street, Bloomington, Indiana 47405

Nikita Selinger
Affiliation: Institute for Mathematical Sciences, Stony Brook University, Stony Brook, New York 11794-3660

DOI: https://doi.org/10.1090/tran/6482
Received by editor(s): April 2, 2013
Received by editor(s) in revised form: May 10, 2014
Published electronically: September 24, 2015
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society