On the completeness of the asymptotic length spectrum Teichmüller space of surfaces of infinite type

AbstractLet X0 be an infinite-type hyperbolic surface (whose boundary components, if any, are closed geodesics) which has an upper bounded pants decomposition. The length spectrum Teichmüller space Tls(X0) consists of all surfaces X homeomorphic to X0 such that the ratios of the corresponding simple closed geodesics are uniformly bounded from below and from above. Alessandrini, Liu, Papadopoulos and Su [1] described the Fenchel–Nielsen coordinates for Tls(X0) and using these coordinates they proved that Tls(X0) is path connected. We use the Fenchel–Nielsen coordinates for Tls(X0) to induce a locally bi-Lipschitz homeomorphism between l∞ and Tls(X0) (which extends analogous results by Fletcher [9] and by Allessandrini, Liu, Papadopoulos, Su and Sun [2] for the unreduced and the reduced Tqc(X0)). Consequently, Tls(X0) is contractible. We also characterize the closure in the length spectrum metric of the quasiconformal Teichmüller space Tqc(X0) in Tls(X0).

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Geodesically Complete Hyperbolic Structures

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004117000792 ◽

2017 ◽

Vol 166 (2) ◽

pp. 219-242

Author(s):

ARA BASMAJIAN ◽

DRAGOMIR ŠARIĆ

Keyword(s):

Teichmüller Space ◽

Hyperbolic Surface ◽

Teichmuller Space ◽

Geometric Proof ◽

Hyperbolic Structures ◽

Infinite Type ◽

Length Spectrum Metric ◽

Geodesically Complete ◽

The Relationship ◽

Convex Core

AbstractIn 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.

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On the inclusion of the quasiconformal Teichmüller space into the length-spectrum Teichmüller space

Monatshefte für Mathematik ◽

10.1007/s00605-015-0813-9 ◽

2015 ◽

Vol 179 (2) ◽

pp. 165-189 ◽

Cited By ~ 3

Author(s):

D. Alessandrini ◽

L. Liu ◽

A. Papadopoulos ◽

W. Su

Keyword(s):

Teichmüller Space ◽

Length Spectrum ◽

Teichmuller Space

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On Geodesic Segments in the Infinitesimal Asymptotic Teichmüller Spaces

Journal of Function Spaces ◽

10.1155/2015/276719 ◽

2015 ◽

Vol 2015 ◽

pp. 1-7

Author(s):

Yan Wu ◽

Yi Qi ◽

Zunwei Fu

Keyword(s):

Banach Space ◽

Riemann Surface ◽

Dual Space ◽

Teichmüller Space ◽

Geodesic Segment ◽

Teichmuller Space ◽

Quadratic Differentials ◽

Teichmüller Spaces ◽

Teichmuller Spaces ◽

Infinite Type

LetAZ(R)be the infinitesimal asymptotic Teichmüller space of a Riemann surfaceRof infinite type. It is known thatAZ(R)is the quotient Banach space of the infinitesimal Teichmüller spaceZ(R), whereZ(R)is the dual space of integrable quadratic differentials. The purpose of this paper is to study the nonuniqueness of geodesic segment joining two points inAZ(R). We prove that there exist infinitely many geodesic segments between the basepoint and every nonsubstantial point in the universal infinitesimal asymptotic Teichmüller spaceAZ(D)by constructing a special degenerating sequence.

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