On local comparison between various metrics on Teichmüller spaces

Abstract

International audienceThere are several Teichmüller spaces associated to a surface of infinite topological type, after the choice of a particular basepoint ( a complex or a hyperbolic structure on the surface). These spaces include the quasiconformal Teichmüller space, the length spectrum Teichmüller space, the Fenchel-Nielsen Teichmüller space, and there are others. In general, these spaces are set-theoretically different. An important question is therefore to understand relations between these spaces. Each of these spaces is equipped with its own metric, and under some hypotheses, there are inclusions between these spaces. In this paper, we obtain local metric comparison results on these inclusions, namely, we show that the inclusions are locally bi-Lipschitz under certain hypotheses. To obtain these results, we use some hyperbolic geometry estimates that give new results also for surfaces of finite type. We recall that in the case of a surface of finite type, all these Teichmüller spaces coincide setwise. In the case of a surface of finite type with no boundary components (and possibly with punctures), we show that the restriction of the identity map to any thick part of Teichmüller space is globally bi-Lipschitz with respect to the length spectrum metric and the classical Teichmüller metric on the domain and on the range respectively. In the case of a surface of finite type with punctures and boundary components, there is a metric on the Teichmüller space which we call the arc metric, whose definition is analogous to the length spectrum metric, but which uses lengths of geodesic arcs instead of lengths of closed geodesics. We show that the restriction of the identity map restricted to any ``relative thick" part of Teichmüller space is globally bi-Lipschitz, with respect to any of the three metrics: the length spectrum metric, the Teichmüller metric and the arc metric on the domain and on the range

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

“…In the same paper, Shiga showed that if the hyperbolic metric R carries a geodesic pants decomposition that satisfies Shiga's condition, then d ls and d qc induce the same topology on T qc (R). In the paper [4] we strengthened this result by showing that under Shiga's condition the inclusion map is locally bi-Lipschitz.…”

“…Then we proved that d qc (X, X i ) → ∞, while d ls (X, X i ) → 0. If we set Y n = τ tn αn • · · · • τ t2 α2 • τ t1 α1 (X) and if we define Y ∞ to be the surface obtained from X by a twist of magnitude t i along α i for every i, then a similar argument shows that Y ∞ ∈ T ls (R) \ T qc (R) and lim n→∞ d ls (Y n , Y ∞ ) = 0; see [3], [4] for more details. We shall give another proof of the last result in Section 5 below.…”

“…The fact that the metric space (T qc (R), d ls ) is, in general, not complete is an indication of the fact that T qc (R) is not the right space for this distance. The length-spectrum Teichmüller space was defined in [12], with the idea that it was the most natural space for that distance, and it was studied in [3], [4]. We proved in [3] that for every base complex structure R, the metric space (T ls (R), d ls ) is complete.…”

The Teichmüller space of the Hirsch foliation

“…In particular, for surfaces of infinite topological type, Teichmüller space can be defined, using either pants decompositions, complex structures, or length-spectrum. But these definitions may yield different spaces as show in [ALP + 11, ALPS12b] and [ALPS12a].…”

On the length spectrum Teichmüller spaces of Riemann surfaces of infinite type

“…For a hyperbolic Riemann surface R 0 , consider a pair (R, f ) of a Riemann surface R and a quasiconformal mapping f from R 0 to R. Such pairs (R 1 , f 1 ) and (R 2 , f 2 We introduce another metric on T (R 0 ). Let C(R 0 ) be the set of non-trivial and non-peripheral closed curves in R 0 .…”