Entropy, Weil-Petersson translation distance and Gromov norm for surface automorphisms

Kojima, Sadayoshi

doi:10.1090/S0002-9939-2012-11250-X

Entropy, Weil-Petersson translation distance and Gromov norm for surface automorphisms
HTML articles powered by AMS MathViewer

by Sadayoshi Kojima PDF

Proc. Amer. Math. Soc. 140 (2012), 3993-4002 Request permission

Abstract:

Thanks to a theorem of Brock on the comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of a Gromov norm of its mapping torus from below and an inbounded geometry case from above. We also prove that the Weil-Petersson translation distance does the same from both sides in general. The proofs are in fact immediately derived from the theorem of Brock, together with some other strong theorems and small observations.

References

R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
Lipman Bers, An extremal problem for quasiconformal mappings and a theorem by Thurston, Acta Math. 141 (1978), no. 1-2, 73–98. MR 477161, DOI 10.1007/BF02545743
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Jeffrey F. Brock, Weil-Petersson translation distance and volumes of mapping tori, Comm. Anal. Geom. 11 (2003), no. 5, 987–999. MR 2032506, DOI 10.4310/CAG.2003.v11.n5.a6
J. Brock, H. Masur and Y. Minsky, Asymptotics of Weil-Petersson geodesics II: Bounded geometry and unbounded entropy, arXiv:math.GT/1004.4401.
Georgios Daskalopoulos and Richard Wentworth, Classification of Weil-Petersson isometries, Amer. J. Math. 125 (2003), no. 4, 941–975. MR 1993745
Travaux de Thurston sur les surfaces, Astérisque, vol. 66, Société Mathématique de France, Paris, 1979 (French). Séminaire Orsay; With an English summary. MR 568308
Frederick P. Gardiner and Nikola Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs, vol. 76, American Mathematical Society, Providence, RI, 2000. MR 1730906, DOI 10.1090/surv/076
E. Kin, S. Kojima, and M. Takasawa, Entropy versus volume for pseudo-Anosovs, Experiment. Math. 18 (2009), no. 4, 397–407. MR 2583541
Michele Linch, A comparison of metrics on Teichmüller space, Proc. Amer. Math. Soc. 43 (1974), 349–352. MR 338453, DOI 10.1090/S0002-9939-1974-0338453-4
Howard Masur, Extension of the Weil-Petersson metric to the boundary of Teichmuller space, Duke Math. J. 43 (1976), no. 3, 623–635. MR 417456
Howard Masur and Michael Wolf, The Weil-Petersson isometry group, Geom. Dedicata 93 (2002), 177–190. MR 1934697, DOI 10.1023/A:1020300413472
Yair N. Minsky, Teichmüller geodesics and ends of hyperbolic $3$-manifolds, Topology 32 (1993), no. 3, 625–647. MR 1231968, DOI 10.1016/0040-9383(93)90013-L
David Mumford, A remark on Mahler’s compactness theorem, Proc. Amer. Math. Soc. 28 (1971), 289–294. MR 276410, DOI 10.1090/S0002-9939-1971-0276410-4
Jean-Pierre Otal, The hyperbolization theorem for fibered 3-manifolds, SMF/AMS Texts and Monographs, vol. 7, American Mathematical Society, Providence, RI; Société Mathématique de France, Paris, 2001. Translated from the 1996 French original by Leslie D. Kay. MR 1855976
G. Perelman, The entropy formula for the Ricci flow and its geometric applications, arXiv:math.DG/0211159.
G. Perelman, Ricci flow with surgery of three-manifolds, arXiv:math.DG/0303109.
Teruhiko Soma, The Gromov invariant of links, Invent. Math. 64 (1981), no. 3, 445–454. MR 632984, DOI 10.1007/BF01389276
W. Thurston, The geometry and topology of $3$-manifolds, Lecture Notes, Princeton University (1979).
William P. Thurston, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 417–431. MR 956596, DOI 10.1090/S0273-0979-1988-15685-6
W. Thurston, Hyperbolic structures on $3$-manifolds II: Surface groups and $3$-manifolds which fiber over the circle, preprint.
Scott Wolpert, Noncompleteness of the Weil-Petersson metric for Teichmüller space, Pacific J. Math. 61 (1975), no. 2, 573–577. MR 422692
Scott A. Wolpert, Chern forms and the Riemann tensor for the moduli space of curves, Invent. Math. 85 (1986), no. 1, 119–145. MR 842050, DOI 10.1007/BF01388794
Scott A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25 (1987), no. 2, 275–296. MR 880186
Scott A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space, Surveys in differential geometry, Vol. VIII (Boston, MA, 2002) Surv. Differ. Geom., vol. 8, Int. Press, Somerville, MA, 2003, pp. 357–393. MR 2039996, DOI 10.4310/SDG.2003.v8.n1.a13