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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Triviality of some representations of $ \operatorname{MCG}(S_g)$ in $ GL(n,\mathbb{C}), \operatorname{Diff}(S^2)$ and $ \operatorname{Homeo}(\mathbb{T}^2)$


Authors: John Franks and Michael Handel
Journal: Proc. Amer. Math. Soc. 141 (2013), 2951-2962
MSC (2010): Primary 20F65, 37E30; Secondary 20F29
Published electronically: May 6, 2013
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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.


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Additional Information

John Franks
Affiliation: Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, Illinois 60208-2730

Michael Handel
Affiliation: Department of Mathematics, Lehman College, 250 Bedford Park Boulevard West, Bronx, New York 10468

DOI: http://dx.doi.org/10.1090/S0002-9939-2013-11556-X
PII: S 0002-9939(2013)11556-X
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
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.