Fenchel-Nielsen coordinates on upper bounded pants decompositions

TL;DR

This paper explores the structure of the length spectrum Teichmüller space for an infinite genus hyperbolic surface with bounded pants decompositions, establishing a homeomorphism with l-infinity and analyzing its topological properties.

Contribution

It extends Fenchel-Nielsen coordinate techniques to the length spectrum Teichmüller space, proving its contractibility and characterizing the closure of the quasiconformal space within it.

Findings

T_{ls}(X_0) is path connected and contractible.

Established a biLipschitz homeomorphism between l-infinity and T_{ls}(X_0).

Characterized the closure of T_{qc}(X_0) in T_{ls}(X_0).

Abstract

Let $X_0$ be an infinite genus hyperbolic surface (whose boundary components, if any, are closed geodesics or punctures) which has an upper bounded pants decomposition. The length spectrum Teichm\"uller space $T_{ls}(X_0)$ consists of all surfaces $X$ homeomorphic to $X_0$ such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su described the Fenchel-Nielsen coordinates for $T_{ls}(X_0)$ and using these coordinates they proved that $T_{ls}(X_0)$ is path connected. We use the Fenchel-Nielsen coordinates for $T_{ls}(X_0)$ to induce a locally biLipschitz homeomorphism between $l^{\infty}$ and $T_{ls}(X_0)$ (which extends analogous results by Fletcher and by Allessandrini, Liu, Papadopoulos, Su and Sun for the unreduced and the reduced $T_{qc}(X_0)$). Consequently, $T_{ls}(X_0)$ is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichm\"uller space $T_{qc}(X_0)$ in $T_{ls}(X_0)$.