Finiteness of mapping class groups and Heegaard distance

2023/03/06 by Yanqing Zou, Zou, Yanqing
Mathematics · #57M50 #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2303.02849

openalex publication_date 2023/03/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that the mapping class group of a Heegaard splitting with a distance of at least 3 is finite. However, we have constructed a counterexample with a distance of 2 that disproves this assertion. In addition, the fact that the mapping class group of a Heegaard splitting with a distance of at most 1 is infinite, when combined with our results, provides an answer to the question of the finiteness of mapping class groups as viewed from Heegaard distance.

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