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Teichmüller displacement theorem on Gromov hyperbolic spaces

Part of: Analysis on metric spaces

Published online by Cambridge University Press:  06 November 2024

Qingshan Zhou Open the ORCID record for Qingshan Zhou [Opens in a new window] ,
Saminathan Ponnusamy Open the ORCID record for Saminathan Ponnusamy [Opens in a new window]  and
Qianghua Luo Open the ORCID record for Qianghua Luo [Opens in a new window]
Qingshan Zhou
Affiliation:
School of Mathematics, Foshan University, Foshan, Guangdong Province, People’s Republic of China
Saminathan Ponnusamy
Affiliation:
Department of Mathematics, Indian Institute of Technology Madras, Chennai, Tamilnadu, India
Qianghua Luo*
Affiliation:
School of Mathematics, Foshan University, Foshan, Guangdong Province, People’s Republic of China
*
Corresponding author: Qianghua Luoemail: luo.qh@fosu.edu.cn
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Abstract

Given a Gromov hyperbolic domain $G\subsetneq \mathbb{R}^n$ with uniformly perfect Gromov boundary, Zhou and Rasila recently proved that for all quasiconformal homeomorphisms $\psi\colon G\to G$ with identity value on the Gromov boundary, the quasihyperbolic displacement $k_G(x,\psi(x))$ for all $x\in G$ is bounded above. In this paper, we generalize this result and establish Teichmüller displacement theorem for quasi-isometries of Gromov hyperbolic spaces in a quantitative way. As applications, we obtain its connections to bilipschitz extensions of certain Gromov hyperbolic spaces.

Keywords

Gromov hyperbolic spaceTeichmüller displacement theoremquasi-isometryroughly starlikeuniformly perfect

MSC classification

Primary: 30L10: Quasiconformal mappings in metric spaces
Secondary: 30L05: Geometric embeddings of metric spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 67 , Issue 4 , November 2024 , pp. 1148 - 1166
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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