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


Author: Dragomir Šarić
Journal: Proc. Amer. Math. Soc. 146 (2018), 2457-2471
MSC (2010): Primary 30F60, 32G15
DOI: https://doi.org/10.1090/proc/13738
Published electronically: March 9, 2018
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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)$.


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Dragomir Šarić
Affiliation: Department of Mathematics, Queens College of CUNY, 65-30 Kissena Boulevard, Flushing, New York 11367 — and — Mathematics PhD Program, The CUNY Graduate Center, 365 Fifth Avenue, New York, New York 10016-4309
Email: Dragomir.Saric@qc.cuny.edu

DOI: https://doi.org/10.1090/proc/13738
Received by editor(s): May 25, 2015
Received by editor(s) in revised form: January 22, 2017
Published electronically: March 9, 2018
Additional Notes: This research was partially supported by National Science Foundation grant DMS 1102440.
Communicated by: Michael Wolf
Article copyright: © Copyright 2018 American Mathematical Society

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