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 Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: For some , let be a finite index subgroup of the mapping class group of a genus surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of should be finite. In the paper we prove two theorems supporting this conjecture. For the first, let denote the Dehn twist about a simple closed curve . For some , we have . We prove that is torsion in the abelianization of . Our second result shows that the abelianization of is finite if 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.

**1.**Joan S. Birman,*Mapping class groups and their relationship to braid groups*, Comm. Pure Appl. Math.**22**(1969), 213–238. MR**0243519****2.**Marco Boggi,*Fundamental groups of moduli stacks of stable curves of compact type*, Geom. Topol.**13**(2009), no. 1, 247–276. MR**2469518**, 10.2140/gt.2009.13.247**3.**M. Bridson, Semisimple actions of mapping class groups on CAT(0) spaces, LMS Lecture Notes, vol. 368,*Geometry of Riemann surfaces*, to appear. http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521733076**4.**Kenneth S. Brown,*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1994. Corrected reprint of the 1982 original. MR**1324339****5.**Richard M. Hain,*Torelli groups and geometry of moduli spaces of curves*, Current topics in complex algebraic geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., vol. 28, Cambridge Univ. Press, Cambridge, 1995, pp. 97–143. MR**1397061****6.**Nikolai V. Ivanov,*Fifteen problems about the mapping class groups*, Problems on mapping class groups and related topics, Proc. Sympos. Pure Math., vol. 74, Amer. Math. Soc., Providence, RI, 2006, pp. 71–80. MR**2264532**, 10.1090/pspum/074/2264532**7.**Dennis Johnson,*An abelian quotient of the mapping class group \cal𝐼_{𝑔}*, Math. Ann.**249**(1980), no. 3, 225–242. MR**579103**, 10.1007/BF01363897**8.**Dennis Johnson,*The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves*, Topology**24**(1985), no. 2, 113–126. MR**793178**, 10.1016/0040-9383(85)90049-7**9.**Dennis Johnson,*The structure of the Torelli group. III. The abelianization of 𝒯*, Topology**24**(1985), no. 2, 127–144. MR**793179**, 10.1016/0040-9383(85)90050-3**10.**Mustafa Korkmaz,*Low-dimensional homology groups of mapping class groups: a survey*, Turkish J. Math.**26**(2002), no. 1, 101–114. MR**1892804****11.**John D. McCarthy,*On the first cohomology group of cofinite subgroups in surface mapping class groups*, Topology**40**(2001), no. 2, 401–418. MR**1808225**, 10.1016/S0040-9383(99)00066-X**12.**Andrew Putman,*Cutting and pasting in the Torelli group*, Geom. Topol.**11**(2007), 829–865. MR**2302503**, 10.2140/gt.2007.11.829**13.**Robert J. Zimmer,*Ergodic theory and semisimple groups*, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. MR**776417**

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

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.