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Geodesically Complete Hyperbolic Structures

Published online by Cambridge University Press:  20 November 2017

ARA BASMAJIAN  and
DRAGOMIR ŠARIĆ
ARA BASMAJIAN
Affiliation:
Ph.D. Program in Mathematics, The Graduate Center of the City University of New York, NY 10016, USA. Department of Mathematics and Statistics, Hunter College, 695 Park Ave., New York, NY 10065, USA. e-mail: [email protected]
DRAGOMIR ŠARIĆ
Affiliation:
Ph.D. Program in Mathematics, The Graduate Center of the City University of New York, NY 10016, USA. Department of Mathematics, Queens College, 65-30 Kissena Blvd., Flushing, NY 11367, USA. e-mail: [email protected]
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Abstract

In the first part of this work we explore the geometry of infinite type surfaces and the relationship between its convex core and space of ends. In particular, we give a geometric proof of a Theorem due to Alvarez and Rodriguez that a geodesically complete hyperbolic surface is made up of its convex core with funnels attached along the simple closed geodesic components and half-planes attached along simple open geodesic components. We next consider gluing infinitely many pairs of pants along their cuffs to obtain an infinite hyperbolic surface. We prove that there always exists a choice of twists in the gluings such that the surface is complete regardless of the size of the cuffs. This generalises the examples of Matsuzaki.

In the second part we consider complete hyperbolic flute surfaces with rapidly increasing cuff lengths and prove that the corresponding quasiconformal Teichmüller space is incomplete in the length spectrum metric. Moreover, we describe the twist coordinates and convergence in terms of the twist coordinates on the closure of the quasiconformal Teichmüller space.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 166 , Issue 2 , March 2019 , pp. 219 - 242
Copyright
Copyright © Cambridge Philosophical Society 2017 

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Footnotes

†

Supported in part by a PSC-CUNY Grant and a Simons foundation grant.

‡

Supported by the National Science Foundation grant DMS 1102440, PSC-CUNY grant, and a Simons foundation grant.

References

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