On local comparison between various metrics on Teichm\"uller spaces

TL;DR

This paper investigates local metric relationships between various Teichmüller spaces associated with surfaces of infinite topological type, establishing bi-Lipschitz equivalences under certain conditions and extending results to finite type surfaces.

Contribution

It provides new local bi-Lipschitz comparison results between different Teichmüller metrics, including length spectrum, Fenchel-Nielsen, and arc metrics, on surfaces of finite and infinite type.

Findings

Inclusions between Teichmüller spaces are locally bi-Lipschitz under certain hypotheses.

Restriction of the identity map to thick parts is globally bi-Lipschitz between length spectrum and classical Teichmüller metrics.

Extension of bi-Lipschitz equivalence to arc metric on surfaces with punctures and boundary components.

Abstract

There are several Teichm\"uller 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 Teichm\"uller space, the length spectrum Teichm\"uller space, the Fenchel-Nielsen Teichm\"uller 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 Teichm\"uller 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 Teichm\"uller space is globally bi-Lipschitz with respect to the length spectrum metric and the classical Teichm\"uller 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 Teichm\"uller 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 Teichm\"uller space is globally bi-Lipschitz, with respect to any of the three metrics: the length spectrum metric, the Teichm\"uller metric and the arc metric on the domain and on the range.