Erina Kinjo | Searcharxiv

arXiv2024

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

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.

math.CV↗

arXiv2022

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

For a generalized Cantor set $E(\omega)$ with respect to a sequence $\omega=\{ q_n \}_{n=1}^{\infty} \subset (0,1)$, we consider Riemann surface $X_{E(\omega)}:=\hat{\mathbb{C}} \setminus E(\omega)$ and metrics on Teichm\"uller space $T(X_{E(\omega)})$ of $X_{E(\omega)}$. If $E(\omega) = \mathcal{C}$ ( the middle one-third Cantor set), we find that on $T(X_{\mathcal{C}})$, Teichm\"uller metric $d_T$ defines the same topology as that of the length spectrum metric $d_L$. Also, we can easily check that $d_T$ does not define the same topology as that of $d_L$ on $T(X_{E(\omega)})$ if $\sup q_n =1$. On the other hand, it is not easy to judge whether the metrics define the same topology or not if $\inf q_n =0$. In this paper, we show that the two metrics define different topologies on $T(X_{E(\omega)})$ for some $\omega=\{ q_n \}_{n=1}^{\infty}$ such that $\inf q_n =0$.

math.CV↗