The Library

Trees and mapping class groups

Tools

Kent, Richard P., Leininger, Christopher J. and Schleimer, Saul. (2009) Trees and mapping class groups. Journal fur die reine und angewandte Mathematik, Vol.2009 (No.637). pp. 1-21. ISSN 0075-4102

Preview

PDF
WRAP_Schleimer_Trees_mapping.pdf - Published Version - Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader
Download (236Kb)

Official URL: http://dx.doi.org/10.1515/CRELLE.2009.087

Abstract

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of B. Farb and L. Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of K. Whittlesey's group. In the course of the proof, we obtain a new proof of a theorem of I. Kra. We also relate the action of this kernel on the curve complex to a family of actions on trees. This quickly yields a new proof of a theorem of J. Harer.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science > Mathematics
Library of Congress Subject Headings (LCSH):	Mappings (Mathematics), Surfaces, Kernel functions
Journal or Publication Title:	Journal fur die reine und angewandte Mathematik
Publisher:	Walter de Gruyter & Co
ISSN:	0075-4102
Date:	December 2009
Volume:	Vol.2009
Number:	No.637
Number of Pages:	21
Page Range:	pp. 1-21
Identification Number:	10.1515/CRELLE.2009.087
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Open Access
Funder:	National Science Foundation (U.S.) (NSF)
Grant number:	DMS-0603881 (NSF), DMS-0508971 (NSF)
References:	[1] Alan F. Beardon, The geometry of discrete groups, Grad. Texts Math. 91, Springer-Verlag, New York 1995. Corrected reprint of the 1983 original. [2] Mladen Bestvina, Questions in geometric group theory, http://www.math.utah.edu/~bestvina. [3] Joan S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213–238. [4] Joan S. Birman, Braids, links, and mapping class groups, Ann. Math. Stud. 82 (1974). [5] Joan S. Birman and Hugh M. Hilden, On the mapping class groups of closed surfaces as covering spaces, in: Advances in the theory of Riemann surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. Math. Stud. 66 (1971), 81–115. [6] Martin R. Bridson and Andre´ Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wiss. 319, Springer-Verlag, Berlin 1999. [7] Max Dehn, Papers on group theory and topology, Springer-Verlag, New York 1987. Translated from the German and with introductions and an appendix by John Stillwell, with an appendix by Otto Schreier. [8] Benson Farb and Lee Mosher, Convex cocompact subgroups of mapping class groups, Geom. Topol. 6 (2002), 91–152. [9] Ursula Hamensta¨dt, Word hyperbolic extensions of surface groups, preprint, arXiv:math.GT/0505244. [10] John L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math. 84(1) (1986), 157–176. [11] Allen Hatcher and Karen Vogtmann, Tethers and homology stability, preprint. [12] Nikolai V. Ivanov, Subgroups of Teichmu¨ ller modular groups, Transl. Math. Monogr. 115, American Mathematical Society, Providence, RI, 1992. Translated from the Russian by E. J. F. Primrose and revised by the author. [13] Richard P. Kent IV and Christopher J. Leininger, Shadows of mapping class groups: capturing convex cocompactness, Geom. Funct. Anal. 18 (2008), 1270–1325. [14] Richard P. Kent IV and Christopher J. Leininger, Subgroups of the mapping class group from the geometrical viewpoint, In the tradition of Ahlfors-Bers, IV, Contemp. Math. 432 (2007), 119–141. [15] Irwin Kra, On the Nielsen-Thurston-Bers type of some self-maps of Riemann surfaces, Acta Math. 146(3–4) (1981), 231–270. [16] Bernard Maskit, Kleinian groups, Grundl. Math. Wiss. 287, Springer-Verlag, Berlin 1988. [17] Lee Mosher, Problems in the geometry of surface group extensions, in: Problems on mapping class groups and related topics, Proc. Sympos. Pure Math. 74 (2006), 245–256. [18] G. Peter Scott and Gadde A. Swarup, Geometric finiteness of certain Kleinian groups, Proc. Amer. Math. Soc. 109(3) (1990), 765–768. [19] Peter B. Shalen, Representations of 3-manifold groups, in: Handbook of geometric topology, North- Holland, Amsterdam (2002), 955–1044. [20] John R. Stallings, Topologically unrealizable automorphisms of free groups, Proc. Amer. Math. Soc. 84(1) (1982), 21–24.
URI:	http://wrap.warwick.ac.uk/id/eprint/16593