Geometric realizations of cyclic actions on surfaces

Abstract

Let Mod(Sg) denote the mapping class group of the closed orientable surface Sg of genus g ≥ 2, and let f ∈ Mod(Sg) be of finite order. We give an inductive procedure to construct an explicit hyperbolic structure on Sg 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(Sg). 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(Sg) → Sp(2g; Z).2000 Mathematics Subject Classification. Primary 57M60; Secondary 57M50, 57M99.

General primitivity in the mapping class group

“…This section discusses the normal generation of Mod(S g ) by a single pseudo-periodic mapping class. Using the theory developed in [19], we obtain a generalization of the result due to Margalit-Lanier [14, Theorem 1.1] for pseudo-periodic mapping classes. In [19], it was shown that an irreducible Type 1 cyclic action can be realized as a rotation of a canonical polygon [19,Theorem 2.7].…”

“…Using the theory developed in [19], we obtain a generalization of the result due to Margalit-Lanier [14, Theorem 1.1] for pseudo-periodic mapping classes. In [19], it was shown that an irreducible Type 1 cyclic action can be realized as a rotation of a canonical polygon [19,Theorem 2.7]. Further, any Type 2 cyclic action can be constructed from certain compatibilities on irreducible Type 1 actions [19,Theorem 2.24].…”

“…Recently, Margalit and Lanier [14] have proved that when g ≥ 3, any periodic mapping class, which is not a hyperelliptic involution, normally generates Mod(S g ). In Section 5, we generalize this result to pseudo-periodic mapping classes by applying the theory developed in [19] and the well-suited curve criterion from [14] (see Proposition 5.2). Keeping in mind that the nontrivial powers of Dehn twists and bounding pair maps do not normally generate Mod(S g ), we show the following as a final application.…”

“…From here on, a periodic mapping class F (and its associated cyclic action) up to conjugacy will be represented by its corresponding data set D F . We will use the following nomenclature (from [19]) for categorizing cyclic actions. ),…”

Metacyclic actions on surfaces

“…In this section, we use Corollary 4.10 to provide explicit geometric realizations of the lifts of some non-split metacyclic actions on S 10 and S 11 . These realizations implicitly assume the theory developed in [4,13]. The associated weak conjugacy classes of these actions are represented by the metacyclic data sets listed in Tables 1-2 in Section 6.…”