Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A note on the abelianizations of finite-index subgroups of the mapping class group


Author: Andrew Putman
Journal: Proc. Amer. Math. Soc. 138 (2010), 753-758
MSC (2000): Primary 57-XX; Secondary 20-XX
Published electronically: September 30, 2009
MathSciNet review: 2557192
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: For some $ g\geq 3$, let $ \Gamma$ be a finite index subgroup of the mapping class group of a genus $ g$ surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $ \Gamma$ should be finite. In the paper we prove two theorems supporting this conjecture. For the first, let $ T_x$ denote the Dehn twist about a simple closed curve $ x$. For some $ n\geq 1$, we have $ T_x^n\in\Gamma$. We prove that $ T_x^n$ is torsion in the abelianization of $ \Gamma$. Our second result shows that the abelianization of $ \Gamma$ is finite if $ \Gamma$ contains a ``large chunk'' (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57-XX, 20-XX

Retrieve articles in all journals with MSC (2000): 57-XX, 20-XX


Additional Information

Andrew Putman
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, 2-306, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139-4307
Email: andyp@math.mit.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-09-10124-7
PII: S 0002-9939(09)10124-7
Received by editor(s): February 3, 2009
Received by editor(s) in revised form: May 19, 2009
Published electronically: September 30, 2009
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.