Thurston's boundary for Teichmüller spaces of infinite surfaces: the length spectrum

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 Teichmüller 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 $PM{L}_{bdd}(X)$, and it coincides with $PM{L}_{bdd}(X)$ 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 $PM{L}_{bdd}(X)$.

arXiv comment13 pages, 2 figures