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

TL;DR

This paper investigates the surjectivity of the symplectic representation of the mapping class group, revealing that it is not surjective when restricted to orientable pseudo-Anosov classes, contrary to the known surjectivity of the full representation.

Contribution

It demonstrates that the symplectic representation's surjectivity fails for orientable pseudo-Anosov classes by explicitly constructing non-representable matrices.

Findings

The full symplectic representation is surjective.

Surjectivity fails when restricted to orientable pseudo-Anosov classes.

Constructed matrices cannot be realized by any orientable pseudo-Anosov mapping class.

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.