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

Abstract

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 P M L bdd (X), and it coincides with P M L bdd (X) when X can be decomposed into a countable union of geodesic pairs of pants whose boundary geodesics {αn} n∈N have lengths pinched between two positive constants. When a subsequence of the lengths of the boundary curves of the geodesic pairs of pants {αn}n converges to zero, Thurston's boundary using the length spectrum is strictly larger than P M L bdd (X).

Limits of Teichmüller geodesics in the Universal Teichmüller space

“…We now establish that α n does not converge to α in the uniform weak* topology if the above conditions are satisfied. (20) and (17)…”

“…Liouville embedding L is in fact a homeomorphism of T (D) onto its image in C(D) equipped with the uniform weak* topology, see [MŠ12,Šar15]. As a set Thurston boundary of T (D), which is denoted by ∂ ∞ T (D), is defined as the collection of asymptotic rays to L (T (D)) in C(D).…”

“…Thurston's boundary to the universal Teichmüller space T (D) is the space of projective bounded measured laminations P M L bdd (D) of D (cf. [18], [20]). We study the limits of Teichmüller geodesic rays on Thurston's boundary to T (D).…”

Train tracks and measured laminations on infinite surfaces

“…It is therefore of interest to understand the space M L b (X) of bounded measured lamination on the hyperbolic surface X. We restrict our attention to surfaces X with bounded pants decompositions (for the definition see [37], [1], [36] and Section 7) where the question is more tractable. We prove (see Theorems 7.4, 8.6 and 9.4) Theorem 1.4.…”