Teichm\"uller spaces of piecewise symmetric homeomorphisms on the unit circle

TL;DR

This paper introduces a new family of Teichmüller spaces, $T_{lat}^X$, interpolating between the universal space and its subspace, based on piecewise symmetric homeomorphisms defined by a subset $X$ of the circle.

Contribution

It defines and studies the properties of $T_{lat}^X$, a novel class of Teichmüller spaces parameterized by subsets of the circle, expanding understanding of their structure and relations.

Findings

$T_{lat}^X$ interpolates between $T$ and $T_0$

Inclusion of $X$ induces inclusion of $T_{lat}^X$

Sequence of $T_{lat}^X$ approximates $T$

Abstract

We interpolate a new family of Teichm\"uller spaces $T_{\sharp}^X$ between the universal Teichm\"uller space $T$ and its little subspace $T_0$, which we call the Teichm\"uller space of piecewise symmetric homeomorphisms. This is defined by prescribing a subset $X$ of the unit circle. The inclusion relation of $X$ induces a natural inclusion of $T_{\sharp}^X$, and an approximation of $T$ is given by an increasing sequence of $T_{\sharp}^X$. In this paper, we discuss the fundamental properties of $T_{\sharp}^X$ from the viewpoint of the quasiconformal theory of Teichm\"uller spaces. We also consider the quotient space of $T$ by $T_{\sharp}^X$ as an analog of the asymptotic Teichm\"uller space.