On countability of Teichmüller modular groups for analytically infinite Riemann surfaces defined by generalized Cantor sets

For any analytically finite Riemann surface, the Teichmüller modular group is countable, but it is not easy to find an analytically infinite Riemann surface for which the Teichmüller modular group is countable. In this paper, we show that the Teichmüller modular group is countable or uncountable for some analytically infinite Riemann surfaces defined by generalized Cantor sets.