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Free subgroups of surface mapping class groups
Authors:
James W. Anderson, Javier Aramayona and Kenneth J. Shackleton
Journal:
Conform. Geom. Dyn. 11 (2007), 44-55
MSC (2000):
Primary 20F65; Secondary 57M50
Posted:
March 15, 2007
Corrigendum:
Conform. Geom. Dyn. 13 (2009), 136-138.
MathSciNet review:
2295997
Full-text PDF Free Access
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Abstract: We quantify the generation of free subgroups of surface mapping class groups by pseudo-Anosov mapping classes in terms of their translation distance and the distance between their axes in Teichmüller's metric. The method makes reference to Teichmüller space only.
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Reine Angew. Math. 473 (1996), 121–136. MR 1390685
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C. Penner, Bounds on least dilatations,
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no. 2, 443–450. MR 1068128
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Igor
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87 (2001), no. 1-3, 345–360. MR 1866856
(2003c:57018), http://dx.doi.org/10.1023/A:1012010721583
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H.
L. Royden, Automorphisms and isometries of Teichmüller
space, Advances in the Theory of Riemann Surfaces (Proc. Conf., Stony
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Princeton, N.J., 1971, pp. 369–383. MR 0288254
(44 #5452)
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William
P. Thurston, Three-dimensional manifolds, Kleinian
groups and hyperbolic geometry, Bull. Amer.
Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524
(83h:57019), http://dx.doi.org/10.1090/S0273-0979-1982-15003-0
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Scott
Wolpert, The length spectra as moduli for compact Riemann
surfaces, Ann. of Math. (2) 109 (1979), no. 2,
323–351. MR
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Scott
A. Wolpert, Geometry of the Weil-Petersson completion of
Teichmüller space, Surveys in differential geometry, Vol. VIII
(Boston, MA, 2002) Surv. Differ. Geom., VIII, Int. Press, Somerville, MA,
2003, pp. 357–393. MR 2039996
(2005h:32032)
- 1.
- J. S. Birman, C. M. Series, Geodesics with bounded intersection number on surfaces are sparsely distributed, Topology 24 (1985), no. 2, 217-225. MR 793185 (87f:57012)
- 2.
- F. Bonahon, Geodesic laminations on surfaces. Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998), 1-37, Contemp. Math., 269. MR 1810534 (2001m:57023)
- 3.
- B. H. Bowditch, Hyperbolic 3-manifolds and the geometry of the curve complex, in ``European Congress of Mathematics, Stockholm, June 27 - July 2, 2004'' (ed. A.Laptev) European Mathematical Society Publishing House (2005) 103-115. MR 2185739 (2006h:57014)
- 4.
- M. R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999. MR 1744486 (2000k:53038)
- 5.
- G. D. Daskalopoulos, L. Katzarkov, R. A. Wentworth, Harmonic maps to Teichmüller space. Math. Res. Lett. 7 (2000), no. 1, 133-146. MR 1748294 (2001b:32022)
- 6.
- G. D. Daskalopoulos, R. A. Wentworth, Classification of Weil-Petersson isometries, Amer. J. Math. 125 (2003) no. 4, 941-975. MR 1993745 (2004d:32011)
- 7.
- B. Farb, L. Mosher, Convex cocompact subgroups of mapping class groups, Geometry and Topology, 6 (2002), 91-152. MR 1914566 (2003i:20069)
- 8.
- H. Hamidi-Tehrani, On free subgroups of the mapping class groups, Preprint (1997).
- 9.
- Y. Imayoshi, M. Taniguchi, An introduction to Teichmüller spaces, Springer-Verlag, Tokyo, 1992. xiv+279 pp. MR 1215481 (94b:32031)
- 10.
- N. V. Ivanov, Coefficients of expansion of pseudo-Anosov homeomorphisms, translation in J. Soviet Math. 52 (1990), no. 1, 2819-2822. MR 964259 (89i:32047)
- 11.
- N. V. Ivanov, Subgroups of Teichmüller modular groups, Translations of Mathematical Monographs 115 (1992). MR 1195787 (93k:57031)
- 12.
- N. V. Ivanov, Mapping class groups, in ``Handbook of geometric topology'' (ed. R. Daverman and R. Sher), Elsevier (2001) 523-633. MR 1886678 (2003h:57022)
- 13.
- R. P. Kent IV, C. J. Leininger, Shadows of mapping class groups: capturing convex co-compactness, arXiv:math.GT/0505114 (2005).
- 14.
- H. A. Masur, On a class of geodesics in Teichmüller space. Ann. of Math. (2) 102 (1975), no. 2, 205-221. MR 0385173 (52:6038)
- 15.
- H. A. Masur, M. Wolf, Teichmüller space is not Gromov hyperbolic, Ann. Acad. Sci. Fenn. Ser. A I Math. 20 (1995), no. 2, 259-267. MR 1346811 (96f:30048)
- 16.
- J. D. McCarthy, A ``Tits-alternative'' for subgroups of surface mapping class groups, Trans. Amer. Math. Soc. 291 (1985), no. 2, 583-612. MR 800253 (87f:57011)
- 17.
- J. D. McCarthy, A. Papadopoulos, Dynamics on Thurston's sphere of projective measured foliations, Comment. Math. Helv. 64 (1989), no. 1, 133-166. MR 982564 (90e:57054)
- 18.
- J. D. McCarthy, A. Papadopoulos, The mapping class group and a theorem of Masur-Wolf. Topology Appl. 96 (1999), no. 1, 75-84. MR 1701241 (2000i:32025)
- 19.
- Y. N. Minsky, Quasi-projections in Teichmüller space. J. Reine Angew. Math. 473 (1996), 121-136. MR 1390685 (97b:32020)
- 20.
- R. C. Penner, Bounds on least dilatations, Proc. Amer. Math. Soc. 113 (1991), no. 2, 443-450. MR 1068128 (91m:57010)
- 21.
- I. Rivin, Simple curves on surfaces. Geom. Dedicata 87 (2001), no. 1-3, 345-360. MR 1866856 (2003c:57018)
- 22.
- H. L. Royden, Automorphisms and isometries of Teichmüller space. 1971, Advances in the Theory of Riemann Surfaces, Ann. of Math. Studies, No. 66. Princeton Univ. Press, Princeton, N.J. MR 0288254 (44:5452)
- 23.
- W. P. Thurston, Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357-381. MR 648524 (83h:57019)
- 24.
- S. A. Wolpert, The length spectra as moduli for compact Riemann surfaces. Ann. of Math. (2) 109 (1979), no. 2, 323-351. MR 528966 (80j:58067)
- 25.
- S. A. Wolpert, Geometry of the Weil-Petersson completion of Teichmüller space. Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), 357-393. MR 2039996 (2005h:32032)
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Additional Information
James W. Anderson
Affiliation:
School of Mathematics, University of Southampton, Southampton SO17 1BJ, England
Email:
j.w.anderson@soton.ac.uk
Javier Aramayona
Affiliation:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, England
Email:
jaram@maths.warwick.ac.uk
Kenneth J. Shackleton
Affiliation:
Institut des Hautes Études Scientifiques, Le Bois-Marie, 35 Route de Chartres, F-91440 Bures-sur-Yvette, France
Address at time of publication:
Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, 2-12-1 O-okayama, Meguro-ku, Tokyo 152-8552, Japan
Email:
kjs2006@alumni.soton.ac.uk; shackleton.k.aa@m.titech.ac.jp
DOI:
http://dx.doi.org/10.1090/S1088-4173-07-00156-7
PII:
S 1088-4173(07)00156-7
Received by editor(s):
May 15, 2006
Received by editor(s) in revised form:
November 8, 2006
Posted:
March 15, 2007
Additional Notes:
The third author was partially supported by a short-term Japan Society for the Promotion of Science post-doctoral fellowship for foreign researchers, number PE05043.
Article copyright:
© Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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