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

Abstract

We work under this hypothesis that the basepoint is upper-bounded and admits short interior curves. There is a natural inclusion of the quasiconformal space in the length-spectrum space. We prove that, under the above hypothesis, the image of this inclusion is nowhere dense in the length-spectrum space. As a corollary we find an explicit description of the length-spectrum Teichmüller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmüller space is pathconnected.

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

“…Allessandrini, Liu, Papadopoulos and Su [2] proved that T (X 0 ) is not complete in the length spectrum metric when there exists a sequence of simple closed geodesics on X 0 whose lengths converge to 0. Thus, X :…”

“…The first surface X 1 that we consider is introduced by Kinjo [13]. Let Γ be the hyperbolic triangle group of signature (2,4,8). Let T be the triangle fundamental polygon for Γ with angles π/2, π/4 and π/8.…”

“…In the case of an infinite surface with a geodesic pants decomposition, the length spectrum metric is defined by (cf. [24], [2])…”

“…The study of the length spectrum properties for infinite surfaces is started by Shiga [24], and it was further developed by various authors(e.g. [1], [2], [3] [15], [13], [18],...).…”

Fenchel-Nielsen coordinates for asymptotically conformal deformations

“…Let X be an infinite hyperbolic surface endowed with an upper bounded geodesic pants decomposition. Alessandrini, Liu, Papadopoulos, Su and Sun [2,3] parametrized the quasiconformal Teichmüller space T qc (X) and the length spectrum Teichmüller space T ls (X) using the Fenchel-Nielsen coordinates. A quasiconformal map f : X → Y is said to be asymptotically conformal if its Beltrami coefficient µ =∂f /∂f converges to zero at infinity.…”