Scholarlink

<p>In this paper, we first prove that any power quasi-symmetry between two metric spaces induces a rough quasi-isometry between their infinite hyperbolic cones. Second, we prove that for a visual Gromov hyperbolic metric space <i>X</i> and a Gromov boundary point <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega \)</EquationSource> </InlineEquation>, <i>X</i> is roughly similar to the infinite hyperbolic cone of its Gromov boundary relative to <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\omega \)</EquationSource> </InlineEquation>. Third, we prove that for a complete metric space <i>Z</i>, there exists a point <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\hat{\omega }\)</EquationSource> </InlineEquation> in the Gromov boundary of its infinite hyperbolic cone such that <i>Z</i> can be seen as the Gromov boundary of its infinite hyperbolic cone relative to <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\hat{\omega }\)</EquationSource> </InlineEquation>. Our results are inspired by Bonk and Schramm (Geom Funct Anal 10:266–306, 2000, Thms. 7.4, 8.1, 8.2).</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Quasi-symmetries Between Metric Spaces and Rough Quasi-isometries Between Their Infinite Hyperbolic Cones

  • 期刊论文

  • 发表日期：

    2026年5月20日

查看全文
  • Manzi Huang,
  • Zhihao Xu

摘要

In this paper, we first prove that any power quasi-symmetry between two metric spaces induces a rough quasi-isometry between their infinite hyperbolic cones. Second, we prove that for a visual Gromov hyperbolic metric space X and a Gromov boundary point \(\omega \) , X is roughly similar to the infinite hyperbolic cone of its Gromov boundary relative to \(\omega \) . Third, we prove that for a complete metric space Z, there exists a point \(\hat{\omega }\) in the Gromov boundary of its infinite hyperbolic cone such that Z can be seen as the Gromov boundary of its infinite hyperbolic cone relative to \(\hat{\omega }\) . Our results are inspired by Bonk and Schramm (Geom Funct Anal 10:266–306, 2000, Thms. 7.4, 8.1, 8.2).