Space of chord-arc curves and BMO/VMO Teichmüller space

Abstract

This paper focuses on the structure of the subspace $T_c$ of the BMO Teichmüller space $T_b$ corresponding to chord-arc curves, which contains the VMO Teichmüller space $T_v$. We prove that $T_c$ is not a subgroup with respect to the group structure of $T_b$, but it is preserved under the inverse operation and the left and the right translations by any element of $T_v$. Moreover, we show that $T_b$ has a fiber structure induced by $T_v$, and the complex structure of $T_b$ can be projected down to the quotient space $T_v\mathrm{\setminus }{T}_b$. Then, we see that $T_c$ consists of fibers of this projection, and its quotient space also has the induced complex structure.