Triviality of some representations of $\operatorname {MCG}(S_g)$ in $GL(n,\mathbb {C}), \operatorname {Diff}(S^2)$ and $\operatorname {Homeo}(\mathbb {T}^2)$
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Abstract:
We show the triviality of representations of the mapping class group of a genus $g$ surface in $GL(n,\mathbb {C}), \operatorname {Diff}(S^2)$ and $\operatorname {Homeo}(\mathbb {T}^2)$ when appropriate restrictions on the genus $g$ and the size of $n$ hold. For example, if $S$ is a surface of finite type with genus $g \ge 3$ and $\phi : \operatorname {MCG}(S) \to GL(n,\mathbb {C})$ is a homomorphism, then $\phi$ is trivial provided $n < 2g.$ We also show that if $S$ is a closed surface with genus $g \ge 7$, then every homomorphism $\phi : \operatorname {MCG}(S) \to \operatorname {Diff}(S^2)$ is trivial and that if $g \ge 3$, then every homomorphism $\phi : \operatorname {MCG}(S)\to \operatorname {Homeo}(\mathbb {T}^2)$ is trivial.References
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Additional Information
- John Franks
- Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730
- MR Author ID: 68865
- Michael Handel
- Affiliation: Department of Mathematics, Lehman College, 250 Bedford Park Boulevard West, Bronx, New York 10468
- MR Author ID: 223960
- Received by editor(s): July 26, 2011
- Received by editor(s) in revised form: November 14, 2011
- Published electronically: May 6, 2013
- Additional Notes: The first author was supported in part by NSF grant DMS0099640.
The second author was supported in part by NSF grant DMS0103435. - Communicated by: Daniel Ruberman
- © Copyright 2013
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 141 (2013), 2951-2962
- MSC (2010): Primary 20F65, 37E30; Secondary 20F29
- DOI: https://doi.org/10.1090/S0002-9939-2013-11556-X
- MathSciNet review: 3068948