#### Vol. 314, No. 2, 2021

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Teichmüller spaces of piecewise symmetric homeomorphisms on the unit circle

### Huaying Wei and Katsuhiko Matsuzaki

Vol. 314 (2021), No. 2, 495–514
##### 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