2020/08/24 by Hyungryul Baik, Baik, Hyungryul, In-Hyeok Choi +3
Mathematics · #Advanced Algebra and Geometry #Advanced Topics in Algebra #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Homotopy and Cohomology in Algebraic Topology
paper · doi:10.48550/arxiv.2008.10142
openalex publication_date 2020/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.