Journal of the American Mathematical Society

Author(s): Mladen Bestvina; Kai-Uwe Bux; Dan Margalit
Journal: J. Amer. Math. Soc. 23 (2010), 61-105.
MSC (2000): Primary 20F34; Secondary 57M07
Posted: July 10, 2009
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Abstract: We prove that the cohomological dimension of the Torelli group for a closed, connected, orientable surface of genus $g \geq 2$ is equal to

. This answers a question of Mess, who proved the lower bound and settled the case of

. We also find the cohomological dimension of the Johnson kernel (the subgroup of the Torelli group generated by Dehn twists about separating curves) to be

. For $g \geq 2$ , we prove that the top dimensional homology of the Torelli group is infinitely generated. Finally, we give a new proof of the theorem of Mess that gives a precise description of the Torelli group in genus 2. The main tool is a new contractible complex, called the ``complex of minimizing cycles'', on which the Torelli group acts.

1.: Problems in low-dimensional topology.
In Rob Kirby, editor, Geometric topology (Athens, GA, 1993), volume 2 of AMS/IP Stud. Adv. Math., pages 35-473. Amer. Math. Soc., Providence, RI, 1997. MR 1470751
2.: Toshiyuki Akita.
Homological infiniteness of Torelli groups.
Topology, 40(2):213-221, 2001. MR 1808217 (2001m:57022)
3.: Mladen Bestvina, Kai-Uwe Bux, and Dan Margalit.
Dimension of the Torelli group for ${Out}(F\sb n)$ .
Invent. Math., 170(1):1-32, 2007. MR 2336078 (2008k:57029)
4.: Robert Bieri.
Groupes à dualité de Poincaré.
C. R. Acad. Sci. Paris Sér. A-B, 273:A6-A8, 1971. MR 0284509 (44:1734)
5.: Joan S. Birman.
On Siegel's modular group.
Math. Ann., 191:59-68, 1971. MR 0280606 (43:6325)
6.: Joan S. Birman.
Braids, links, and mapping class groups.
Princeton University Press, Princeton, N.J., 1974.
Annals of Mathematics Studies, No. 82. MR 0375281 (51:11477)
7.: Joan S. Birman, Alex Lubotzky, and John McCarthy.
Abelian and solvable subgroups of the mapping class groups.
Duke Math. J., 50(4):1107-1120, 1983. MR 726319 (85k:20126)
8.: A. Borel and J.-P. Serre.
Corners and arithmetic groups.
Comment. Math. Helv., 48:436-491, 1973.
Avec un appendice: Arrondissement des variétés à coins, par A. Douady et L. Hérault. MR 0387495 (52:8337)
9.: Kenneth S. Brown.
Cohomology of groups, volume 87 of Graduate Texts in Mathematics.
Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
10.: Peter Buser.
Geometry and spectra of compact Riemann surfaces, volume 106 of Progress in Mathematics.
Birkhäuser Boston Inc., Boston, MA, 1992. MR 1183224 (93g:58149)
11.: Thomas Church.
Orbits of curves under the Johnson kernel.
In preparation.
12.: Thomas Church.
Personal communication.
13.: 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. MR 881797 (88d:01041)
14.: Samuel Eilenberg and Tudor Ganea.
On the Lusternik-Schnirelmann category of abstract groups.
Ann. of Math. (2), 65:517-518, 1957. MR 0085510 (19:52d)
15.: Benson Farb.
Some problems on mapping class groups and moduli space.
In Problems on mapping class groups and related topics, volume 74 of Proc. Sympos. Pure Math., pages 11-55. Amer. Math. Soc., Providence, RI, 2006. MR 2264130 (2007h:57018)
16.: A. T. Fomenko, D. B. Fuchs, and V. L. Gutenmacher.
Homotopic topology.
Akadémiai Kiadó (Publishing House of the Hungarian Academy of Sciences), Budapest, 1986.
Translated from the Russian by K. Mályusz. MR 873943 (88f:55001)
17.: Richard Hain.
The rational cohomology ring of the moduli space of abelian 3-folds.
Math. Res. Lett., 9(4):473-491, 2002. MR 1928867 (2003h:14068)
18.: John L. Harer.
Stability of the homology of the mapping class groups of orientable surfaces.
Ann. of Math. (2), 121(2):215-249, 1985. MR 786348 (87f:57009)
19.: John L. Harer.
The virtual cohomological dimension of the mapping class group of an orientable surface.
Invent. Math., 84(1):157-176, 1986. MR 830043 (87c:32030)
20.: Allen Hatcher.
Spectral sequences.
Preliminary version, available at http://www.math. cornell.edu/ $\sim$ hatcher/SSAT/SSATpage.html.
21.: Allen Hatcher.
The cyclic cycle complex of a surface.
arXiv:0806.0326.
22.: Allen Hatcher.
On triangulations of surfaces.
Topology Appl., 40(2):189-194, 1991. MR 1123262 (92f:57020)
23.: Allen Hatcher.
Algebraic topology.
Cambridge University Press, Cambridge, 2002. MR 1867354 (2002k:55001)
24.: Y. Imayoshi and M. Taniguchi.
An introduction to Teichmüller spaces.
Springer-Verlag, Tokyo, 1992.
Translated and revised from the Japanese by the authors. MR 1215481 (94b:32031)
25.: N. V. Ivanov.
Complexes of curves and Teichmüller spaces.
Mat. Zametki, 49(5):54-61, 158, 1991. MR 1137173 (93a:32032)
26.: Nikolai V. Ivanov.
Mapping class groups.
In Handbook of geometric topology, pages 523-633. North-Holland, Amsterdam, 2002. MR 1886678 (2003h:57022)
27.: Dennis Johnson.
An abelian quotient of the mapping class group ${\mathcal I}\sb{g}$ .
Math. Ann., 249(3):225-242, 1980. MR 579103 (82a:57008)
28.: Dennis Johnson.
Conjugacy relations in subgroups of the mapping class group and a group-theoretic description of the Rochlin invariant.
Math. Ann., 249(3):243-263, 1980. MR 579104 (82a:57007)
29.: Dennis Johnson.
The structure of the Torelli group. I. A finite set of generators for ${\mathcal I}$ .
Ann. of Math. (2), 118(3):423-442, 1983. MR 727699 (85a:57005)
30.: Dennis Johnson.
The structure of the Torelli group. II. A characterization of the group generated by twists on bounding curves.
Topology, 24(2):113-126, 1985. MR 793178 (86i:57011)
31.: F. E. A. Johnson and C. T. C. Wall.
On groups satisfying Poincaré duality.
Ann. of Math. (2), 96:592-598, 1972. MR 0311796 (47:358)
32.: John P. Labute.
On the descending central series of groups with a single defining relation.
J. Algebra, 14:16-23, 1970. MR 0251111 (40:4342)
33.: Wilhelm Magnus, Abraham Karrass, and Donald Solitar.
Combinatorial group theory.
Dover Publications Inc., Mineola, NY, second edition, 2004.
Presentations of groups in terms of generators and relations. MR 2109550 (2005h:20052)
34.: Darryl McCullough and Andy Miller.
The genus Torelli group is not finitely generated.
Topology Appl., 22(1):43-49, 1986. MR 831180 (87h:57015)
35.: Greg McShane and Igor Rivin.
A norm on homology of surfaces and counting simple geodesics.
Internat. Math. Res. Notices, (2):61-69 (electronic), 1995. MR 1317643 (96b:57014)
36.: Geoffrey Mess.
Unit tangent bundle subgroups of the mapping class groups.
Preprint IHES/M/90/30, 1990.
37.: Geoffrey Mess.
The Torelli groups for genus and surfaces.
Topology, 31(4):775-790, 1992. MR 1191379 (93k:57003)
38.: Luis Paris and Dale Rolfsen.
Geometric subgroups of mapping class groups.
J. Reine Angew. Math., 521:47-83, 2000. MR 1752295 (2001b:57035)
39.: Andrew Putman.
The Johnson homomorphism and its kernel.
Preprint. arXiv:0904.0467.
40.: Andrew Putman.
Cutting and pasting in the Torelli groups.
Geom. Topol., 11:829-865 (electronic), 2007. MR 2302503 (2008c:57049)
41.: Jean-Pierre Serre.
Cohomologie des groupes discrets.
In Prospects in mathematics (Proc. Sympos., Princeton Univ., Princeton, N.J., 1970), pages 77-169. Ann. of Math. Studies, No. 70. Princeton Univ. Press, Princeton, N.J., 1971. MR 0385006 (52:5876)
42.: Jean-Pierre Serre.
Trees.
Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.
Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation. MR 607504 (82c:20083)
43.: John R. Stallings.
On torsion-free groups with infinitely many ends.
Ann. of Math. (2), 88:312-334, 1968. MR 0228573 (37:4153)
44.: Richard G. Swan.
Groups of cohomological dimension one.
J. Algebra, 12:585-610, 1969. MR 0240177 (39:1531)
45.: William P. Thurston.
A norm for the homology of -manifolds.
Mem. Amer. Math. Soc., 59(339):i-vi and 99-130, 1986. MR 823443 (88h:57014)
46.: William R. Vautaw.
Abelian subgroups of the Torelli group.
Algebr. Geom. Topol., 2:157-170 (electronic), 2002. MR 1917048 (2003g:57027)

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Mladen Bestvina
Affiliation: Department of Mathematics, University of Utah, 155 S 1400 East, Salt Lake City, Utah 84112-0090
Email: bestvina@math.utah.edu

Kai-Uwe Bux
Affiliation: Department of Mathematics, University of Virginia, Kerchof Hall 229, Charlottesville, Virginia 22903-4137
Email: kb2ue@virginia.edu

Dan Margalit
Affiliation: Department of Mathematics, Tufts University, 503 Boston Avenue, Medford, Massachusetts 02155
Email: dan.margalit@tufts.edu

DOI: 10.1090/S0894-0347-09-00643-2
PII: S 0894-0347(09)00643-2
Keywords: Mapping class group, Torelli group, Johnson kernel, cohomological dimension, complex of minimizing cycles
Received by editor(s): September 7, 2007
Posted: July 10, 2009
Additional Notes: The first and third authors gratefully acknowledge support by the National Science Foundation.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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ISSN: 1088-6834(e) ISSN: 0894-0347(p)

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