Determining Thurston classes using Nielsen types
Author: Jane Gilman
Journal: Trans. Amer. Math. Soc. 272 (1982), 669-675
MSC: Primary 57N05; Secondary 30F99, 32G15, 57N05
MathSciNet review: 662059
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Abstract: In previous work  we showed how the Thurston or Bers classifications of diffeomorphisms of surfaces could be obtained using the Nielsen types of the lifts of the diffeomorphism to the unit disc. In this paper we find improved conditions on the Nielsen types for the Thurston and Bers classes. We use them to verify that an example studied by Nielsen is a pseudo-Anosov diffeomorphism with stretching factor of degree 4. This example is of interest in its own right, but it also serves to illustrate exactly how the Nielsen types are used for verifying examples. We discuss the general usefulness of this method.
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L. Bers, An extremal problem for quasiconformal mapping and a theorem of Thurston, Acta Math. 141 (1978), 73-98.
J. Franks, unpublished.
J. Gilman, On the Nielsen type and the classification for the mapping-class group, Adv. in Math. 40 (1981), 68-96.
J. Hubbard and H. Masur, Quadratic differentials and foliations, Acta Math. 142 (1979), 223-274.
R. Miller, Nielsen’s viewpoint on geodesic laminations, Adv. in Math. (to appear).
J. Nielsen, Surface transformation classes of algebraically finite type, Mat.-Fys. Medd. Danske Vid. Selsk. 21 (2) (1944), 1-89.
---, Untersuchungen zur Topologie der geschlossenen zweiseitigen Flachen. Parts I-III, Acta Math. 50 (1927), 189-358; 53 (1929), 1-76; 58 (1932), 87-167.
V. Poénaru et al., Traveaux de Thurston, Astérisque 66-67, Société Mathématique de France, 1980.
W. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces. I, preprint.
W. Veech, Gauss measures of transformations on the space of interval exchange maps, Ann. of Math. (to appear).
Keywords: Mapping-class group, Teichmüller modular group, Riemann surface, automorphism, pseudo-Anosov diffeomorphism, foliations, Fuchsian group
Article copyright: © Copyright 1982 American Mathematical Society