A note on the abelianizations of finite-index subgroups of the mapping class group. (English) Zbl 1207.57006
Author’s 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.
Reviewer: Masako Kobayashi (Osaka)
MSC:
| 57M07 | Topological methods in group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 57M99 | General low-dimensional topology |
References:
| [1] | Joan S. Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969), 213 – 238. · Zbl 0167.21503 · doi:10.1002/cpa.3160220206 |
| [2] | Marco Boggi, Fundamental groups of moduli stacks of stable curves of compact type, Geom. Topol. 13 (2009), no. 1, 247 – 276. · Zbl 1162.32008 · doi: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. |
| [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. · Zbl 0868.14006 |
| [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. · Zbl 1281.57011 · doi:10.1090/pspum/074/2264532 |
| [7] | Dennis Johnson, An abelian quotient of the mapping class group \cal\?_{\?}, Math. Ann. 249 (1980), no. 3, 225 – 242. · Zbl 0409.57009 · doi: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. · Zbl 0571.57009 · doi:10.1016/0040-9383(85)90049-7 |
| [9] | Dennis Johnson, The structure of the Torelli group. III. The abelianization of \?, Topology 24 (1985), no. 2, 127 – 144. · Zbl 0571.57010 · doi: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. · Zbl 1026.57015 |
| [11] | John D. McCarthy, On the first cohomology group of cofinite subgroups in surface mapping class groups, Topology 40 (2001), no. 2, 401 – 418. · Zbl 0976.14023 · doi:10.1016/S0040-9383(99)00066-X |
| [12] | Andrew Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007), 829 – 865. · Zbl 1157.57010 · doi:10.2140/gt.2007.11.829 |
| [13] | Robert J. Zimmer, Ergodic theory and semisimple groups, Monographs in Mathematics, vol. 81, Birkhäuser Verlag, Basel, 1984. · Zbl 0571.58015 |
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