Geometry & Topology

All finite groups are involved in the mapping class group

Gregor Masbaum and Alan W Reid

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Let Γg denote the orientation-preserving mapping class group of the genus g1 closed orientable surface. In this paper we show that for fixed g, every finite group occurs as a quotient of a finite index subgroup of Γg.

Article information

Geom. Topol., Volume 16, Number 3 (2012), 1393-1411.

Received: 22 September 2011
Revised: 11 May 2012
Accepted: 5 March 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F38: Other groups related to topology or analysis
Secondary: 57R56: Topological quantum field theories

mapping class group finite quotient representation Zariski dense subgroup


Masbaum, Gregor; Reid, Alan W. All finite groups are involved in the mapping class group. Geom. Topol. 16 (2012), no. 3, 1393--1411. doi:10.2140/gt.2012.16.1393.

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  • I Agol, D D Long, A W Reid, The Bianchi groups are separable on geometrically finite subgroups, Ann. of Math. 153 (2001) 599–621
  • H Bass, J Milnor, J-P Serre, Solution of the congruence subgroup problem for ${\rm SL}_{n}\,(n\geq 3)$ and ${\rm Sp}_{2n}\,(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. (1967) 59–137
  • N Bergeron, F Haglund, D T Wise, Hyperplane sections in arithmetic hyperbolic manifolds, J. Lond. Math. Soc. 83 (2011) 431–448
  • J A Berrick, V Gebhardt, L Paris, Finite index subgroups of mapping class groups
  • C Blanchet, N Habegger, G Masbaum, P Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995) 883–927
  • A Borel, Compact Clifford–Klein forms of symmetric spaces, Topology 2 (1963) 111–122
  • A Borel, Linear algebraic groups, W. A. Benjamin, New York-Amsterdam (1969) Notes taken by H Bass
  • A Borel, Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. 75 (1962) 485–535
  • A Borel, J Tits, Compléments à l'article: “Groupes réductifs”, Inst. Hautes Études Sci. Publ. Math. (1972) 253–276
  • N M Dunfield, W P Thurston, Finite covers of random $3$–manifolds, Invent. Math. 166 (2006) 457–521
  • N M Dunfield, H Wong, Quantum invariants of random $3$–manifolds, Algebr. Geom. Topol. 11 (2011) 2191–2205
  • L Funar, Zariski density and finite quotients of mapping class groups, to appear in Int. Math. Res. Not. 2012 (2012)
  • L Funar, W Pitsch, Finite quotients of symplectic groups vs mapping class groups
  • R Gilman, Finite quotients of the automorphism group of a free group, Canad. J. Math. 29 (1977) 541–551
  • P M Gilmer, G Masbaum, Maslov index, Lagrangians, mapping class groups and TQFT, to appear in Forum Math.
  • P M Gilmer, G Masbaum, Integral lattices in TQFT, Ann. Sci. École Norm. Sup. 40 (2007) 815–844
  • P M Gilmer, G Masbaum, P van Wamelen, Integral bases for TQFT modules and unimodular representations of mapping class groups, Comment. Math. Helv. 79 (2004) 260–284
  • E K Grossman, On the residual finiteness of certain mapping class groups, J. London Math. Soc. 9 (1974/75) 160–164
  • F Grunewald, A Lubotzky, Linear representations of the automorphism group of a free group, Geom. Funct. Anal. 18 (2009) 1564–1608
  • D Johnson, The structure of the Torelli group I: A finite set of generators for ${\mathcal I}$, Ann. of Math. 118 (1983) 423–442
  • M Korkmaz, On cofinite subgroups of mapping class groups, Turkish J. Math. 27 (2003) 115–123
  • M Larsen, Z Wang, Density of the $\mathrm{SO}(3)$ TQFT representation of mapping class groups, Comm. Math. Phys. 260 (2005) 641–658
  • C J Leininger, D B McReynolds, Separable subgroups of mapping class groups, Topology Appl. 154 (2007) 1–10
  • D D Long, A W Reid, Surface subgroups and subgroup separability in $3$–manifold topology, IMPA Math. Publ., Inst. Nacional de Mat. Pura e Aplicada, Rio de Janeiro (2005) 25th Brazilian Math. Colloquium
  • E Looijenga, Prym representations of mapping class groups, Geom. Dedicata 64 (1997) 69–83
  • A Lubotzky, D Segal, Subgroup growth, Progress in Math. 212, Birkhäuser, Basel (2003)
  • G Masbaum, On representations of mapping class groups in Integral TQFT, Oberwolfach Reports 5 (2008) 1157–1232 Available at \setbox0\makeatletter\@url {\unhbox0
  • G Masbaum, J D Roberts, On central extensions of mapping class groups, Math. Ann. 302 (1995) 131–150
  • G Mess, The Torelli groups for genus $2$ and $3$ surfaces, Topology 31 (1992) 775–790
  • D W Morris, Ratner's theorems on unipotent flows, Chicago Lectures in Math., Univ. of Chicago Press (2005)
  • G D Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980) 171–276
  • M V Nori, On subgroups of ${\rm GL}_n({\bf F}_p)$, Invent. Math. 88 (1987) 257–275
  • V Platonov, A Rapinchuk, Algebraic groups and number theory, Pure and Applied Math. 139, Academic Press, Boston (1994) Translated from the 1991 Russian original by R Rowen
  • N Reshetikhin, V G Turaev, Invariants of $3$–manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991) 547–597
  • G Shimura, Arithmetic of unitary groups, Ann. of Math. 79 (1964) 369–409
  • G Shimura, Arithmetic of Hermitian forms, Doc. Math. 13 (2008) 739–774
  • J Tits, Classification of algebraic semisimple groups, from: “Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, CO, 1965)”, (A Borel, G D Mostow, editors), Amer. Math. Soc. (1966) 33–62
  • J Tits, Reductive groups over local fields, from: “Automorphic forms, representations and $L$–functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, OR, 1977), Part 1”, (A Borel, editor), Proc. Sympos. Pure Math. XXXIII, Amer. Math. Soc. (1979) 29–69
  • E B Vinberg, Rings of definition of dense subgroups of semisimple linear groups, Izv. Akad. Nauk SSSR Ser. Mat. 35 (1971) 45–55
  • B Weisfeiler, Strong approximation for Zariski-dense subgroups of semisimple algebraic groups, Ann. of Math. 120 (1984) 271–315