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Papers by Athanase Papadopoulos

Monatshefte Fur Mathematik, 2008

This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped wi... more This paper has two parts. In the first part, we study shift coordinates on a sphere S equipped with three distinguished points and a triangulation whose vertices are the distinguished points. These coordinates parametrize a space $\widetilde{{\cal T}}(S)$ that we call an unfolded Teichmüller space. This space contains Teichmüller spaces of the sphere with ${\frak b}$ boundary components and ${\frak p}$ cusps (which we call generalized pairs of pants), for all possible values of ${\frak b}$ and ${\frak p}$ satisfying ${\frak b}+{\frak p}=3$ . The parametrization of $\widetilde{{\cal T}}(S)$ by shift coordinates equips this space with a natural polyhedral structure, which we describe more precisely as a cone over an octahedron in ${\Bbb {R}}^3$ . Each cone over a simplex of this octahedron is interpreted as a Teichmüller space of the sphere with ${\frak b}$ boundary components and ${\frak p}$ cusps, for fixed ${\frak b}$ and ${\frak p}$ , the sphere being furthermore equipped with an orientation on each boundary component. There is a natural linear action of a finite group on $\widetilde{{\cal T}}(S)$ whose quotient is an augmented Teichmüller space in the usual sense. We describe several aspects of the geometry of the space $\widetilde{{\cal T}}(S)$ . Stretch lines and earthquakes can be defined on this space. In the second part of the paper, we use the shift coordinates to obtain estimates on the behaviour of stretch lines in the Teichmüller space of a surface obtained by gluing hyperbolic pairs of pants. We also use the shift coordinates to give formulae that express stretch lines in terms of Fenchel-Nielsen coordinates. We deduce the disjointness of some stretch lines in Teichmüller space. We study in more detail the case of a closed surface of genus 2.

$Research paper thumbnail of Some metrics on Teichm\"uller spaces of surfaces of infinite type$

Unlike the case of surfaces of topologically finite type, there are several different Teichm\"ull... more Unlike the case of surfaces of topologically finite type, there are several different Teichm\"uller spaces that are associated to a surface of topological infinite type. These Teichm\"uller spaces first depend (set-theoretically) on whether we work in the hyperbolic category or in the conformal category. They also depend, given the choice of a point of view (hyperbolic or conformal), on the choice of a distance function on Teichm\"uller space. Examples of distance functions that appear naturally in the hyperbolic setting are the length spectrum distance and the bi-Lipschitz distance, and there are other useful distance functions. The Teichm\"uller spaces also depend on the choice of a basepoint. The aim of this paper is to present some examples, results and questions on the Teichm\"uller theory of surfaces of infinite topological type that do not appear in the setting the Teichm\"uller theory of surfaces of finite type. In particular, we point out relations and differences between the various Teichm\"uller spaces associated to a given surface of topological infinite type.

Monatshefte Fur Mathematik, 2010

We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite... more We consider some metrics and weak metrics defined on the Teichmüller space of a surface of finite type with nonempty boundary, that are defined using the hyperbolic length spectrum of simple closed curves and of properly embedded arcs, and we compare these metrics and weak metrics with the Teichmüller metric. The comparison is on subsets of Teichmüller space which we call “ε 0-relative ${\epsilon}$ -thick parts”, and whose definition depends on the choice of some positive constants ε 0 and ${\epsilon}$ . Meanwhile, we give a formula for the Teichmüller metric of a surface with boundary in terms of extremal lengths of families of arcs.

Annales Academiae Scientiarum Fennicae-mathematica, 2010

We define and study natural metrics and weak metrics on the Teichmüller space of a surface of top... more We define and study natural metrics and weak metrics on the Teichmüller space of a surface of topologically finite type with boundary. These metrics and weak metrics are associated to the hyperbolic length spectrum of simple closed curves and of properly embedded arcs in the surface. We give a comparison between the defined metrics on regions of Teichmüller space which we call ε 0 -relative -thick parts, for > 0 and ε 0 ≥ > 0. We compare the topologies defined by these metrics on Teichmüller space and we study divergence to infinity with respect to these various metrics.

Mathematical Proceedings of The Cambridge Philosophical Society, 2007

In this paper, we establish some properties of Thurston's asymmetric metric L on the Teichmüller ... more In this paper, we establish some properties of Thurston's asymmetric metric L on the Teichmüller space T g,n of a surface of genus g with n punctures and with negative Euler characteristic. We study convergence of sequences of elements in T g,n in the sense of L, as well as sequences that tend to infinity in T g,n . We show that the topology that the asymmetric metric L induces on Teichmüller space is the same as the usual topology. Furthermore, we show that L satisfies the axioms of a (not necessarily symmetric) metric in the sense of Busemann and conclude that L is complete in the sense of Busemann.

Proceedings of The American Mathematical Society, 2009

We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on ... more We give a proof of an unpublished result of Thurston showing that given any hyperbolic metric on a surface of finite type with nonempty boundary, there exists another hyperbolic metric on the same surface for which the lengths of all simple closed geodesics are shorter. (This is not possible for surfaces of finite type with empty boundary.) Furthermore, we show that we can do the shortening in such a way that it is bounded below by a positive constant. This improves a recent result obtained by Parlier in . We include this result in a discussion of the weak metric theory of the Teichmüller space of surfaces with nonempty boundary.

| Let X be a CAT(?1)? space which is spherically symmetric around some point x 0 2 X and whose bo... more | Let X be a CAT(?1)? space which is spherically symmetric around some point x 0 2 X and whose boundary has nite positive s?dimensional Hausdor measure. Let = ( x ) x2X be a conformal density of dimension d > s=2 on @X. We prove that x 0 is a weak limit of measures supported on spheres centered at x 0 . These measures are expressed in terms of the total mass function of and of the d?dimensional spherical function on X. In particular, this result proves that is entirely determined by its dimension and its total mass function. The results of this paper apply in particular for symmetric spaces of rank one and semi-homogeneous trees.

Manuscripta Mathematica, 2002

Let Γ be a word hyperbolic group M. Gromov has constructed a compact space equipped with a flow ... more Let Γ be a word hyperbolic group M. Gromov has constructed a compact space equipped with a flow which is defined up to orbit-equivalence and which is called the geodesic flow of Γ. In the special case where Γ is the fundamental group of a Riemannian manifold of negative sectional curvature, is the unit tangent bundle of the manifold equipped with the usual geodesic flow. In this paper, we construct, for every hyperbolic group Γ, a subshift of finite type and a continuous map from the suspension of this subshift onto , which is uniformly bounded-to-one and which sends each orbit of the suspension flow onto an orbit of the geodesic flow.

Glasgow Mathematical Journal, 2001

Let X be a proper geodesic metric space which is -hyperbolic in the sense of Gromov. We study a c... more Let X be a proper geodesic metric space which is -hyperbolic in the sense of Gromov. We study a class of functions on X, called horofunctions, which generalize Busemann functions. To each horofunction is associated a point in the boundary at infinity of X. Horofunctions are used to give a description of the boundary. In the case where X is the Cayley graph of a hyperbolic group À, we show, following ideas of Gromov sketched in his paper Hyperbolic groups, that the space of cocycles associated to horofunctions which take integral values on the vertices is a one-sided subshift of finite type.

Journal of The London Mathematical Society-second Series, 1997

Mathematical Proceedings of The Cambridge Philosophical Society, 1997

Let G be a connected locally finite simplicial graph with rk([pi]1(G))[gt-or-equal, slanted]2 and... more Let G be a connected locally finite simplicial graph with rk([pi]1(G))[gt-or-equal, slanted]2 and let T be the universal cover of G. Consider a [pi]1(G)-invariant conformal density [mu] of dimension d on [partial partial differential]T. The total mass function [phi][mu] of [mu] is defined on the set of vertices of G. Let |[phi][mu]| be its l2-norm. Let [Omega] be the geodesic

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Mathematische Annalen, 1988