On the surjectivity of the symplectic representation of the mapping class group (Q2105032)

Let \(S_g\) be a closed connected orientable surface of genus \(g\). The mapping class group \(\mathrm{MCG}(S_g)\) of \(S_g\) is the group of orientation-preserving homeomorphisms of \(S\), up to isotopy. The symplectic representation \[ \Psi \colon \mathrm{MCG}(S_g) \rightarrow \mathrm{Sp}(2g, \mathbb{Z}) \] records the action of a mapping class on the first homology group \(H_1(S_g; \mathbb{Z})\) equipped with the algebraic intersection pairing. It is well-known that the symplectic representation is surjective. It is also surjective after restricting further to pseudo-Anosov elements of the mapping class group. It is natural to ask about the image of the symplectic representation upon restricting further to certain subsets of the mapping class group, for example orientable pseudo-Anosov mapping classes (i.e. those pseudo-Anosov elements that have an orientable invariant foliation). If \(\phi\) is an orientable pseudo-Anosov element, then the following are known about \(A = \Psi(\phi)\) \begin{itemize} \item[1.] The leading (i.e. largest in absolute value) eigenvalue of \(A\) is equal to the stretch factor \(\lambda\) of \(\phi\). In particular, the leading eigenvalue of \(A\) is a biPerron algebraic integer \(\lambda\), meaning that all other roots of the minimal polynomial of \(\lambda\) lie in the annulus \(\{ z \in \mathbb{C} \colon 1/\lambda < |z| < \lambda \}\) except possibly for \(1/\lambda\) or \(- 1/ \lambda\). Note that the definition of a biPerron number in this article is weaker than the one given in this review. However, it is well-known that stretch factors of pseudo-Anosov maps on \textit{connected} surfaces satisfy this stronger condition. See Theorem 1 of \textit{D. Fried} [Ergodic Theory Dyn. Syst. 5, 539--563 (1985; Zbl 0603.58020)]. \item[2.] \(\lambda\) is a simple root of the characteristic polynomial of \(A\); i.e. of algebraic multiplicity one. \end{itemize} The authors give examples of symplectic matrices in \(\mathrm{Sp}(2g, \mathbb{Z})\) with biPerron leading eigenvalue such that no root of the characteristic polynomial is a simple root. It follows that the second condition is not a consequence of the first condition. In particular, there are symplectic matrices with biPerron leading eigenvalue which cannot be obtained as the symplectic representation of an orientable pseudo-Anosov mapping class. A more natural question is whether every symplectic matrix satisfying both conditions 1 and 2 is realised as the image of an orientable pseudo-Anosov mapping class. However, given that Fried's surface entropy conjecture is widely open at the time of this writing, it is not clear how this question could be approached.