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

2015/05/25 by Saric, Dragomir
#FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1505.06692

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 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 \α_n\_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 \α_n\_n converges to zero, Thurston's boundary using the length spectrum is strictly larger than PML_bdd(X).

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