Geometric realizations of cyclic actions on surfaces

2017/05/29 by Shiv Parsad, Kashyap Rajeevsarathy, Parsad, Shiv +3 · 1 citation
Mathematics · #57M99 #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Primary: 57M60 #Secondary: 57M50

paper · pdf · doi:10.48550/arxiv.1705.10206

openalex publication_date 2017/05/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Mod(S_g) denote the mapping class group of the closed orientable surface S_g of genus g≥ 2, and let f∈ Mod(S_g) be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on S_g that realizes f as an isometry. In other words, this procedure yields an explicit solution to the Nielsen realization problem for cyclic subgroups of Mod(S_g). Furthermore, we give a purely combinatorial perspective by showing how certain finite order mapping classes can be viewed as fat graph automorphisms. As an application of our realizations, we determine the sizes of maximal reduction systems for certain finite order mapping classes. Moreover, we describe a method to compute the image of finite order mapping classes and the roots of Dehn twists, under the symplectic representation Ψ: Mod(S_g) → Sp(2g; ℤ).

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• Fenchel-Nielsen coordinates of the branch loci of cyclic actions

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