On the surjectivity of the symplectic representation of the mapping class group

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In this note, we study the symplectic representation of the mapping class group. In particular, we discuss the surjecivity of the representation restricted to certain mapping classes. It is well-known that the representation itself is surjective. In fact the representation is still surjective after restricting on pseudo-Anosov mapping classes. However, we show that the surjectivity does not hold when the representation is restricted on orientable pseudo-Anosovs, even after reducing its codomain to integer symplectic matrices with a bi-Perron leading eigenvalue. In order to prove the non-surjectivity, we explicitly construct an infinite family of symplectic matrices with a bi-Perron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class. © 2022 Elsevier B.V.
Publisher
Elsevier B.V.
Issue Date
2022-12
Language
English
Article Type
Article
Citation

TOPOLOGY AND ITS APPLICATIONS, v.322

ISSN
0166-8641
DOI
10.1016/j.topol.2022.108334
URI
http://hdl.handle.net/10203/303480
Appears in Collection
MA-Journal Papers(저널논문)
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