Thurston's boundary for Teichm\"uller 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 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 $PML_{bdd}(X)$.