The horofunction compactification of Teichm\"uller metric

We show that the horofunction compactification of Teichm\"uller space with the Teichm\"uller metric is homeomorphic to the Gardiner-Masur compactification.Comment: Final version, the proof of Proposition 5.1 is improved. To appear in Handbook of Teichm\"uller Theory, Vol. 4, A. Papadopoulos (ed.), EMS publishing house, Z\"urich 201

Thurston's metric on Teichm\"uller space and the translation lengths of mapping classes

We show that the Teichm\"uller space of a surface without boundary and with punctures, equipped with Thurston's metric is the limit (in an appropriate sense) of Teichm\"uller spaces of surfaces with boundary, equipped with their arc metrics, when the boundary lengths tend to zero. We use this to obtain a result on the translation distances for mapping classes for their actions on Teichm\"uller spaces equipped with their arc metrics

On hyperbolic analogues of some classical theorems in spherical geometry

We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise

A comparison between the Avila-Gou\"ezel-Yoccoz norm and the Teichm\"uller norm

We give a comparison between the Avila-Gou\"ezel-Yoccoz norm and the Teichm\"uller norm on the principal stratum of holomorphic quadratic differentials.Comment: 10 pages,2 figur

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 Teichm\"uller 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 Teichm\"uller 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