Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T12:22:40.119Z Has data issue: false hasContentIssue false
  • English
  • Français

Factoring in the hyperelliptic Torelli group†

Published online by Cambridge University Press:  19 June 2015

TARA E. BRENDLE  and
DAN MARGALIT
TARA E. BRENDLE
Affiliation:
School of Mathematics and Statistics, University of Glasgow, 15 University Gardens, Glasgow G12 8QW, e-mail: tara.brendle@glasgow.ac.uk
DAN MARGALIT
Affiliation:
School of Mathematics, Georgia Institute of Technology, 686 Cherry Street, Atlanta, GA 30332U.S.A. e-mail: margalit@math.gatech.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The hyperelliptic Torelli group is the subgroup of the mapping class group consisting of elements that act trivially on the homology of the surface and that also commute with some fixed hyperelliptic involution. Putman and the authors proved that this group is generated by Dehn twists about separating curves fixed by the hyperelliptic involution. In this paper, we introduce an algorithmic approach to factoring a wide class of elements of the hyperelliptic Torelli group into such Dehn twists, and apply our methods to several basic types of elements. As one consequence, we answer an old question of Dennis Johnson.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 159 , Issue 2 , September 2015 , pp. 207 - 217
Copyright
Copyright © Cambridge Philosophical Society 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

Supported in part by EPSRC grant EP/J019593/1. Supported by the Sloan Foundation and the NSF Grant No. DMS - 1057874.

References

REFERENCES

[1] Artin, E. Theory of braids. Ann. of Math. (2) 48 (1947), 101–126.Google Scholar
[2] Brendle, T. and Margalit, D. Factoring in the hyperelliptic Torelli group (version 1). arXiv:1202.2365v1.Google Scholar
[3] Brendle, T. and Margalit, D. The level four braid group. J. reine angew. Math., to appear.Google Scholar
[4] Brendle, T., Margalit, D. and Putman, A. Generators for the hyperelliptic Torelli group and the kernel of the Burau representation at t = -1. Invent. Math., 200 (2015), no. 1, 263–310.Google Scholar
[5] Brendle, T. E. and Margalit, D. Point pushing, homology, and the hyperelliptic involution. Michigan Math. J. 62 (3) (2013), 451–473.Google Scholar
[6] Farb, B. and Margalit, D. A Primer on Mapping Class Groups. Princeton University Press, 2011.Google Scholar
[7] Gonzàlez-Meneses, J. Basic results on braid groups. arXiv:1010.0321.Google Scholar
[8] Hain, R. Finiteness and Torelli spaces. In Problems on mapping class groups and related topics. Proc. Sympos. Pure Math. vol. 74 (Amer. Math. Soc., Providence, RI, 2006), pp. 57–70.Google Scholar
[9] Putman, A. A note on the connectivity of certain complexes associated to surfaces. Enseign. Math. (2) 54 (3–4) 2008, 287–301.Google Scholar