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

Abstract

Abstract. On the Teichmüller space T (R 0 ) of a hyperbolic Riemann surface R 0 , we consider the length spectrum metric d L , which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if R 0 is of finite type, then d L defines the same topology as that of Teichmüller metric d T on T (R 0 ). In 2003, H. Shiga extended the discussion to the Teichmüller spaces of Riemann surfaces of infinite type and proved that the two metrics define the same topology on T (R 0 ) if R 0 satisfies some geometric condition. After that, Alessandrini-Liu-Papadopoulos-Su proved that for the Riemann surface satisfying Shiga's condition, the identity map between the two metric spaces is locally bi-Lipschitz.In this paper, we extend their results; that is, we show that if R 0 has bounded geometry, then the identity map (T (R 0 ), d L ) → (T (R 0 ), d T ) is locally bi-Lipschitz.

Bounded geometry with no bounded pants decomposition

“…It is clear that a Riemann surface with a bounded pants decomposition has bounded geometry; however, the converse is false-Kinjo gives a counterexample in [9]. (However, Kinjo shows that every hyperbolic Riemann surface with bounded geometry does admit a bounded hexagonal decomposition [10].) In fact, the Riemann surface R obtained by removing the lattice Z ⊕ iZ from C is such a counterexample (which is distinct from Kinjo's example, but similar in nature).…”

On the length spectrums of Riemann surfaces given by generalized Cantor sets

“…All Riemann surfaces of finite type and some Riemann surfaces of infinite type admit such a decomposition. After that Liu-Sun-Wei ( [6]; 2008) and Kinjo ([2]; 2011, [3]; 2014, [4]; 2018) gave sufficient conditions for the two metrics to define the same topology or different ones.…”