A note on the abelianizations of finiteindex subgroups of the mapping class group
Author:
Andrew Putman
Journal:
Proc. Amer. Math. Soc. 138 (2010), 753758
MSC (2000):
Primary 57XX; Secondary 20XX
Published electronically:
September 30, 2009
MathSciNet review:
2557192
Fulltext 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
(39 #4840)
 2.
Marco
Boggi, Fundamental groups of moduli stacks of stable curves of
compact type, Geom. Topol. 13 (2009), no. 1,
247–276. MR 2469518
(2010c:14027), 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, SpringerVerlag, New York, 1994. Corrected reprint of the
1982 original. MR 1324339
(96a:20072)
 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
(97d:14036)
 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
(2008b:57003), 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
(82a:57008), 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
(86i:57011), 10.1016/00409383(85)900497
 9.
Dennis
Johnson, The structure of the Torelli group. III. The
abelianization of 𝒯, Topology 24 (1985),
no. 2, 127–144. MR 793179
(87a:57016), 10.1016/00409383(85)900503
 10.
Mustafa
Korkmaz, Lowdimensional homology groups of mapping class groups: a
survey, Turkish J. Math. 26 (2002), no. 1,
101–114. MR 1892804
(2003f:57002)
 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
(2001m:57029), 10.1016/S00409383(99)00066X
 12.
Andrew
Putman, Cutting and pasting in the Torelli group, Geom. Topol.
11 (2007), 829–865. MR 2302503
(2008c:57049), 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
(86j:22014)
 1.
 J. S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213238. MR 0243519 (39:4840)
 2.
 M. Boggi, Fundamental groups of moduli stacks of stable curves of compact type, Geom. Topol. 13 (2009), 247276. MR 2469518
 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.
 K. S. Brown, Cohomology of groups, corrected reprint of the 1982 original, SpringerVerlag, New York, 1994. MR 1324339 (96a:20072)
 5.
 R. M. Hain, Torelli groups and geometry of moduli spaces of curves, in Current topics in complex algebraic geometry (Berkeley, CA, 1992/93), 97143, Cambridge Univ. Press, Cambridge, 1995. MR 1397061 (97d:14036)
 6.
 N. V. Ivanov, Fifteen problems about the mapping class groups, in Problems on mapping class groups and related topics, 7180, Proc. Sympos. Pure Math., 74, Amer. Math. Soc., Providence, RI, 2006. MR 2264532 (2008b:57003)
 7.
 D. Johnson, An abelian quotient of the mapping class group , Math. Ann. 249 (1980), no. 3, 225242. MR 579103 (82a:57008)
 8.
 D. Johnson, The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves, Topology 24 (1985), no. 2, 113126. MR 793178 (86i:57011)
 9.
 D. Johnson, The structure of the Torelli group. III. The abelianization of , Topology 24 (1985), no. 2, 127144. MR 793179 (87a:57016)
 10.
 M. Korkmaz, Lowdimensional homology groups of mapping class groups: A survey, Turkish J. Math. 26 (2002), no. 1, 101114. MR 1892804 (2003f:57002)
 11.
 J. D. McCarthy, On the first cohomology group of cofinite subgroups in surface mapping class groups, Topology 40 (2001), no. 2, 401418. MR 1808225 (2001m:57029)
 12.
 A. Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007), 829865. MR 2302503 (2008c:57049)
 13.
 R. J. Zimmer, Ergodic theory and semisimple groups, Birkhäuser, Basel, 1984. MR 776417 (86j:22014)
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
57XX,
20XX
Retrieve articles in all journals
with MSC (2000):
57XX,
20XX
Additional Information
Andrew Putman
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, 2306, 77 Massachusetts Avenue, Cambridge, Massachusetts 021394307
Email:
andyp@math.mit.edu
DOI:
http://dx.doi.org/10.1090/S0002993909101247
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.
