# Geometric realizations of cyclic actions on surfaces

## TL;DR

This paper provides an explicit geometric construction for finite order mapping classes on surfaces, solving the Nielsen realization problem for cyclic subgroups and linking combinatorial and algebraic perspectives.

## Contribution

It introduces an inductive method to realize cyclic mapping classes as isometries on hyperbolic surfaces and connects these to fat graph automorphisms and symplectic representations.

## Key findings

- Explicit hyperbolic structures for cyclic actions are constructed.
- Finite order mapping classes are characterized as fat graph automorphisms.
- Methods to compute images and roots of Dehn twists under symplectic representation.

## Abstract

Let $ \text{Mod}(S_g)$ denote the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$, and let $f\in \text{Mod}(S_g)$ 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 $ \text{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 $\Psi: \text{Mod}(S_g) \to \text{Sp}(2g; \mathbb{Z})$.