Estimating the distances between hyperbolic structures in the moduli space

TL;DR

This paper develops a method to estimate distances between fixed points of cyclic actions on Teichmüller space of surfaces, using pants decompositions and the pants graph to analyze hyperbolic structures.

Contribution

It introduces a novel approach combining pants decompositions and quasi-isometry to estimate distances between fixed points in Teichmüller space.

Findings

Explicit description of pants decompositions with bounded curve lengths

Establishment of a quasi-isometry-based distance estimation method

Application to fixed points of cyclic subgroup actions

Abstract

Let $\mathrm{Mod}(S_g)$ be the mapping class group of the closed orientable surface $S_g$ of genus $g\geq 2$. Given a finite subgroup $H$ of $\mathrm{Mod}(S_g)$, let $\mathrm{Fix}(H)$ be the set of all fixed points induced by the action of $H$ on the Teichm\"{u}ller space $\mathrm{Teich}(S_g)$ of $S_g$. This paper provides a method to estimate the distance between the unique fixed points of certain irreducible cyclic actions on $S_g$. We begin by deriving an explicit description of a pants decomposition of $S_g$, the length of whose curves are bounded above by the Bers' constant. To obtain the estimate, our method then uses the quasi-isometry between $\mathrm{Teich}(S_g)$ and the pants graph $\mathcal{P}(S_g)$.