It is an open question whether mapping class groups of a closed orientable surface is linear. One natural representation of a mapping class group into a linear group is its symplectic representation, which is induced from the action of the mapping class group on the first homology of the surface. It is well-known that the symplectic representation is surjective onto the integral lattice.
It is a natural question that what we can observe when we restrict the symplectic representation on the set of pseudo-Anosov mapping classes. Thurston proved that there are plenty of pseudo-Anosovs in the kernel of the symplectic representation, employing his construction of pseudo-Anosovs from filling multicurves.
In this talk, we show that the surjectivity still holds after restricting the symplectic representation on the set of pseudo-Anosovs. On the other hand, we also show that the surjectivity does not hold on the set of pseudo-Anosovs with orientable invariant measured foliations. This is the joint work with Hyungryul Baik and Inhyeok Choi, answering the question of Ursula Hamenstädt.