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arXiv · 1908.08798

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

Huaying Wei·Katsuhiko Matsuzaki
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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.

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Huaying Wei, Katsuhiko Matsuzaki. 2019-08-22. Teichm\"uller spaces of piecewise symmetric homeomorphisms on the unit circle. https://doi.org/10.2140/pjm.2021.314.495

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