Teichmüller spaces of piecewise symmetric homeomorphisms on the unit circle

Huaying Wei and Katsuhiko Matsuzaki

Vol. 314 (2021), No. 2, 495–514

DOI: 10.2140/pjm.2021.314.495

Abstract

We interpolate a new family of Teichmüller spaces $T_{♯}^X$ between the universal Teichmüller space $T$ and its little subspace $T_0$. Each $T_{♯}^X$ is defined by prescribing a subset $X$ of the unit circle as the exceptional set of the vanishing property for $T_0$. The inclusion relation of $X$ induces a natural inclusion of $T_{♯}^X$, and an approximation of $T$ by an increasing sequence of $T_{♯}^X$ is investigated. In this paper, we discuss the fundamental properties of $T_{♯}^X$ from the viewpoint of the quasiconformal theory of Teichmüller spaces. We also consider the quotient space of $T$ by $T_{♯}^X$ as an analog of the asymptotic Teichmüller space.

Keywords

universal Teichmüller space, symmetric homeomorphism, asymptotically conformal, Bers embedding, barycentric extension

Mathematical Subject Classification 2010

Primary: 30C62, 30F60, 32G15

Secondary: 37E10, 58D05

Milestones

Received: 26 December 2019

Revised: 16 December 2020

Accepted: 14 August 2021

Published: 10 November 2021