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

Abstract:

Let $X_0$ be an infinite area geodesically complete hyperbolic surface which can be decomposed into geodesic pairs of pants. We introduce Thurston’s boundary to the Teichmüller space $T(X_0)$ of the surface $X_0$ using the length spectrum analogous to Thurston’s construction for finite surfaces. Thurston’s boundary using the length spectrum is a “closure” of projective bounded measured laminations $PML_{bdd} (X_0)$, and it coincides with $PML_{bdd}(X_0)$ when $X_0$ 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 $PML_{bdd}(X_0)$.