The horofunction compactification of the arc metric on Teichmuller space

The arc metric is an asymmetric metric on the Teichm{u}ller space T(S) of a surface S with nonempty boundary. In this paper we study the relation between Thurstons compactification and the horofunction compactification of T(S) endowed with the arc metric. We prove that there is a natural homeomorphism between the two compactifications.

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

Given a surface of infinite topological type, there are several Teichmuller spaces associated with it, depending on the basepoint and on the point of view that one uses to compare different complex structures. This paper is about the comparison between the quasiconformal Teichmuller space and the length-spectrum Teichmuller space. 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 Teichmuller space in terms of Fenchel-Nielsen coordinates and we prove that the length-spectrum Teichmuller space is path-connected.

On the classification of mapping class actions on Thurstons asymmetric metric

We study the action of the elements of the mapping class group of a surface of finite type on the Teichmuller space of that surface equipped with Thurstons asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurstons classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmuller space equipped with Teichmullers metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmuller space equipped with the Weil-Petersson metric.

The behaviour of Fenchel-Nielsen distance under a change of pants decomposition

Given a topological orientable surface of finite or infinite type equipped with a pair of pants decomposition $mathcal{P}$ and given a base complex structure $X$ on $S$, there is an associated deformation space of complex structures on $S$, which we call the Fenchel-Nielsen Teichmuller space associated to the pair $(mathcal{P},X)$. This space carries a metric, which we call the Fenchel-Nielsen metric, defined using Fenchel-Nielsen coordinates. We studied this metric in the papers cite{ALPSS}, cite{various} and cite{local}, and we compared it to the classical Teichmuller metric (defined using quasi-conformal mappings) and to another metric, namely, the length spectrum, defined using ratios of hyperbolic lengths of simple closed curves metric. In the present paper, we show that under a change of pair of pants decomposition, the identity map between the corresponding Fenchel-Nielsen metrics is not necessarily bi-Lipschitz. The results complement results obtained in the previous papers and they show that these previous results are optimal.