Thurston's boundary for Teichm\"uller spaces of infinite surfaces: the length spectrum

TL;DR

This paper extends Thurston's boundary concept to infinite hyperbolic surfaces using the length spectrum, revealing conditions under which the boundary coincides with or exceeds the space of bounded measured laminations.

Contribution

It introduces a length spectrum-based Thurston boundary for infinite surfaces and characterizes when it matches or surpasses bounded measured laminations.

Findings

Thurston's boundary is a closure of projective bounded measured laminations.

Boundary coincides with PML_bdd(X) when boundary lengths are uniformly bounded.

Boundary is larger when boundary lengths tend to zero in subsequences.

Abstract

Let $X$ be an infinite geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston's boundary to the Teichm\"uller space $T(X)$ of the surface $X$ using the length spectrum analogous to Thurston's construction for finite surfaces. Thurston's boundary using the length spectrum of $X$ is a "closure" of projective bounded measured laminations $PML_{bdd} (X)$, and it coincides with $PML_{bdd}(X)$ when $X$ can be decomposed into a countable union of geodesic pairs of pants whose boundary geodesics $\{\alpha_n\}_{n\in\mathbb{N}}$ have lengths pinched between two positive constants. When a subsequence of the lengths of the boundary curves of the geodesic pairs of pants $\{\alpha_n\}_n$ converges to zero, Thurston's boundary using the length spectrum is strictly larger than $PML_{bdd}(X)$.