On countability of Teichm\"uller modular groups for analytically infinite Riemann surfaces defined by generalized Cantor sets

TL;DR

This paper investigates the countability of Teichmüller modular groups for analytically infinite Riemann surfaces constructed via generalized Cantor sets, revealing conditions under which these groups are countable or uncountable.

Contribution

It provides new insights into the countability properties of Teichmüller modular groups for a class of analytically infinite Riemann surfaces.

Findings

Teichmüller modular group is countable for some surfaces.

Teichmüller modular group is uncountable for other surfaces.

The results depend on the structure of the generalized Cantor sets.

Abstract

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