Subgroups generated by two pseudo-Anosov elements in a mapping class group. II. Uniform bound on exponents

2009/08/07 by Koji Fujiwara, Fujiwara, Koji
Computer Science · Mathematics · #20F65 #20F67 #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #math.GR #math.GT #msc:20F65 #msc:20F67 #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.0908.0995

33 pages, 11 figures

arxiv created 2009/08/07 · openalex publication_date 2009/08/07 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let S be a compact orientable surface, and \Mod(S) its mapping class group. Then there exists a constant M(S), which depends on S, with the following property. Suppose a,b ∈ \Mod(S) are independent (i.e., [aⁿ,b^m]\not=1 for any n,m \not=0) pseudo-Anosov elements. Then for any n,m ≥ M, the subgroup <aⁿ,b^m> is free of rank two, and convex-cocompact in the sense of Farb-Mosher. In particular all non-trivial elements in <aⁿ,b^m> are pseudo-Anosov. We also show that there exists a constant N, which depends on a,b, such that <aⁿ,b^m> is free of rank two and convex-cocompact if |n|+|m| ≥ N and nm \not=0.

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