2013/12/20 by Thomas Koberda, Koberda, Thomas, Johanna Mangahas +1
Computer Science · Mathematics · #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1312.6141
openalex publication_date 2013/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article, we propose two algorithms for determining the Nielsen-Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in ⟨ X ⟩. We show that if ψ represents a reducible mapping class in \Mod(S) then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use this fact to find the canonical reduction system of ψ. We also prove an effective conjugacy separability result for π1(S) which allows us to lift the action of ψ to a finite cover \ytS of S whose degree depends computably on the word length of ψ, and to use the homology action of ψ on H1(\ytS,ℂ) to determine the Nielsen-Thurston classification of ψ.