Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

An algebraic formulation of Thurston's combinatorial equivalence


Author: Kevin M. Pilgrim
Journal: Proc. Amer. Math. Soc. 131 (2003), 3527-3534
MSC (2000): Primary 37F20; Secondary 20F28, 20F36, 20E05
DOI: https://doi.org/10.1090/S0002-9939-03-07035-7
Published electronically: May 7, 2003
MathSciNet review: 1991765
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $f:S^2 \to S^2$ be an orientation-preserving branched covering for which the set $P_f$ of strict forward orbits of critical points is finite and let $G=\pi_1(S^2-f^{-1}P_f)$. To $f$ we associate an injective endomorphism $\varphi_f$ of the free group $G$, well-defined up to postcomposition with inner automorphisms. We show that two such maps $f,g$ are combinatorially equivalent (in the sense introduced by Thurston for the characterization of rational functions as dynamical systems) if and only if $\varphi_f, \varphi_g$ are conjugate by an element of $\operatorname{Out}(G)$ which is induced by an orientation-preserving homeomorphism.


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

  • [B] P. Brinkmann.
    Hyperbolic automorphisms of free groups.
    Geom. Funct. Anal. 10(2000), no. 5, 1071-1089. MR 2001m:20061
  • [DV] Warren Dicks and Enric Ventura.
    The group fixed by a family of injective endomorphisms of a free group.
    American Mathematical Society, Providence, RI, 1996. MR 97h:20030
  • [DH] A. Douady and John Hubbard.
    A proof of Thurston's topological characterization of rational functions.
    Acta. Math. 171(1993), 263-297. MR 94j:58143
  • [FH] Michael Handel and Mark Feighn.
    Mapping tori of free group automorphisms are coherent.
    Ann. of Math. (2) 149(1999), no. 3, 1061-1077. MR 2000i:20050
  • [GMSW] Ross Geoghegan, Michael L. Mihalik, Mark Sapir, and Daniel T. Wise.
    Ascending HNN extensions of finitely generated free groups are Hopfian.
    Bull. London Math. Soc. 33(2001), no. 3, 292-298. MR 2002a:20029
  • [Kap] Ilya Kapovich.
    Mapping tori of endomorphisms of free groups.
    Comm. Algebra 28(2000), 2895-2917. MR 2001c:20098
  • [Kam] Atsushi Kameyama.
    The Thurston equivalence for postcritically finite branched coverings.
    Osaka J. Math. 38(2001), 565-610. MR 2002h:57004
  • [Lev] Silvio Levy.
    Critically Finite Rational Maps.
    Ph.D. thesis, Princeton University, 1986.
  • [Pil1] Kevin M. Pilgrim.
    Dessins d'enfants and Hubbard trees.
    Ann. Sci. École Norm. Sup. (4) 33(2000), 671-693. MR 2002m:37062
  • [Pil2] Kevin M. Pilgrim.
    Canonical Thurston obstructions.
    Adv. Math. 158(2001), 154-168. MR 2001m:57004
  • [ST] Mitsuhiro Shishikura and Tan Lei.
    A family of cubic rational maps and matings of cubic polynomials.
    Experiment. Math. 9(2000), 29-53. MR 2001c:37042
  • [Sta] John R. Stallings.
    Topology of finite graphs
    Invent. Math. 71(1983), 551-565. MR 85m:05037a
  • [Tan] Tan Lei.
    Matings of quadratic polynomials.
    Ergodic Theory Dynamical Systems 12(1992), 589-620. MR 93h:58129
  • [ZVC] Heiner Zieschang, Elmar Vogt, and Hans-Dieter Coldewey.
    Surfaces and planar discontinuous groups.
    Springer, Berlin, 1980.
    Translated from the German by John Stillwell. MR 82h:57002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 37F20, 20F28, 20F36, 20E05

Retrieve articles in all journals with MSC (2000): 37F20, 20F28, 20F36, 20E05


Additional Information

Kevin M. Pilgrim
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405-7106
Email: pilgrim@indiana.edu

DOI: https://doi.org/10.1090/S0002-9939-03-07035-7
Keywords: Postcritically finite, endomorphism of free group
Received by editor(s): June 20, 2002
Published electronically: May 7, 2003
Additional Notes: This research was supported by Indiana University, Bloomington
Communicated by: Michael Handel
Article copyright: © Copyright 2003 American Mathematical Society

American Mathematical Society