On the surjectivity of the Symplectic representation of the mapping class group
Hyungryul Baik, Inhyeok Choi, Dongryul M. Kim
TL;DR
The paper investigates whether the symplectic representation $\Psi$ of the mapping class group remains surjective when restricted to orientable pseudo-Anosovs onto the subset of symplectic matrices with bi-Perron leading eigenvalue. It leverages the known surjectivity of $\Psi$ on the full group and Fried's result that stretch factors of pseudo-Anosovs are bi-Perron, together with McMullen's result that the leading eigenvalue is simple for orientable pseudo-Anosovs. The main result is a constructive negative answer: for each genus $g \ge 2$, there exist $A \in \mathrm{Sp}(2g,\mathbb{Z})$ with a bi-Perron leading eigenvalue but no simple eigenvalues, hence not arising as $\Psi(\varphi)$ for any orientable pseudo-Anosov, yielding an infinite family of obstructions. Consequently, while $\Psi$ is surjective on pseudo-Anosovs, it fails to be onto the bi-Perron-leading-eigenvalue class when restricted to orientable pseudo-Anosovs, informing the broader discussion related to Fried's conjecture.
Abstract
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.