Estimating the distances between hyperbolic structures in the moduli space
Atreyee Bhattacharya, Suman Paul, Kashyap Rajeevsarathy
TL;DR
The paper tackles the problem of bounding Weil-Petersson distances between fixed-point submanifolds Fix$(H)$ in Teichmüller space for finite cyclic subgroups of the mapping class group. It develops explicit admissible pants decompositions for irreducible Type 1 actions by leveraging hyperbolic polygons $\mathcal{P}_F$, Bers' constant, and the data-set framework that encodes periodic conjugacy classes, then translates geometric questions into pants-graph combinatorics via Brock's quasi-isometry. A key contribution is a combinatorial encoding of pants decompositions as canonical $2g$-tuples, along with an explicit distance bound in the moduli space: $\hat{d}_{WP}([\mathcal{P}_{F_1}],[\mathcal{P}_{F_2}])\leq K\mathbb{D}([f_1],[f_2])+\epsilon$, where $\mathbb{D}$ measures combinatorial divergence of the associated building blocks. The results bridge hyperbolic geometry, orbifold data, and combinatorial topology of the pants graph, enabling computable coarse-distance estimates between totally geodesic submanifolds in $(\mathrm{Teich}(S_g),\mu_{wp})$.
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 Teichmüller 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 $\mathcal{P}(S_g)$.