Accidental parabolics in the mapping class group

Leininger, Christopher

doi:10.1090/S0002-9939-08-09604-4

Accidental parabolics in the mapping class group
HTML articles powered by AMS MathViewer

by Christopher J. Leininger PDF

Proc. Amer. Math. Soc. 137 (2009), 1153-1160 Request permission

Abstract:

In this paper we discuss the behavior of the Gromov boundaries and limit sets for the surface subgroups of the mapping class group with accidental parabolics constructed by the author and A. Reid (2006). Specifically, we show that generically there are no Cannon–Thurston maps from the Gromov boundary to Thurston’s boundary of Teichmüller space.

References

Danny Calegari, Real places and torus bundles, Geom. Dedicata 118 (2006), 209–227. MR 2239457, DOI 10.1007/s10711-005-9037-9
Travaux de Thurston sur les surfaces, Société Mathématique de France, Paris, 1991 (French). Séminaire Orsay; Reprint of Travaux de Thurston sur les surfaces, Soc. Math. France, Paris, 1979 [ MR0568308 (82m:57003)]; Astérisque No. 66-67 (1991) (1991). MR 1134426
William J. Floyd, Group completions and limit sets of Kleinian groups, Invent. Math. 57 (1980), no. 3, 205–218. MR 568933, DOI 10.1007/BF01418926
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, DOI 10.1090/mmono/115
Richard P. Kent IV and Christopher J. Leininger, Subgroups of mapping class groups from the geometrical viewpoint, In the tradition of Ahlfors-Bers. IV, Contemp. Math., vol. 432, Amer. Math. Soc., Providence, RI, 2007, pp. 119–141. MR 2342811, DOI 10.1090/conm/432/08306
C. J. Leininger, Graphs of Veech groups, work in progress.
Christopher J. Leininger, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer’s number, Geom. Topol. 8 (2004), 1301–1359. MR 2119298, DOI 10.2140/gt.2004.8.1301
C. J. Leininger and A. W. Reid, A combination theorem for Veech subgroups of the mapping class group, Geom. Funct. Anal. 16 (2006), no. 2, 403–436. MR 2231468, DOI 10.1007/s00039-006-0556-9
Bernard Maskit, Kleinian groups, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 287, Springer-Verlag, Berlin, 1988. MR 959135
Howard Masur, Two boundaries of Teichmüller space, Duke Math. J. 49 (1982), no. 1, 183–190. MR 650376
John McCarthy and Athanase Papadopoulos, Dynamics on Thurston’s sphere of projective measured foliations, Comment. Math. Helv. 64 (1989), no. 1, 133–166. MR 982564, DOI 10.1007/BF02564666
Lee Mosher, Problems in the geometry of surface group extensions, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 245–256. MR 2264544, DOI 10.1090/pspum/074/2264544
R. C. Penner and J. L. Harer, Combinatorics of train tracks, Annals of Mathematics Studies, vol. 125, Princeton University Press, Princeton, NJ, 1992. MR 1144770, DOI 10.1515/9781400882458
W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. MR 1005006, DOI 10.1007/BF01388890
W. A. Veech, Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards, Invent. Math. 97 (1989), no. 3, 553–583. MR 1005006, DOI 10.1007/BF01388890