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

Abstract

On the Teichmüller space T ( R 0 ) T(R_0) of a hyperbolic Riemann surface R 0 R_0 , we consider the length spectrum metric d L d_L , which measures the difference of hyperbolic structures of Riemann surfaces. It is known that if R 0 R_0 is of finite type, then d L d_L defines the same topology as that of Teichmüller metric d T d_T on T ( R 0 ) 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 ) T(R_0) if R 0 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 R_0 has bounded geometry, then the identity map ( T ( R 0 ) , d L ) → ( T ( R 0 ) , d T ) (T(R_0),d_L) \to (T(R_0),d_T) is locally bi-Lipschitz.