Scholarlink

<p>Given a group <i>G</i>, its poset of hyperbolic structures <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal {H}(G)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> encodes all the possible cobounded actions of <i>G</i> on hyperbolic spaces. In this article, we describe the poset <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {H}(H_n)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">H</mi> <mo stretchy="false">(</mo> <msub> <mi>H</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for every Houghton group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(H_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(n \ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. In particular, we show that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(H_n\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>H</mi> <mi>n</mi> </msub> </math></EquationSource> </InlineEquation> admits exactly <i>n</i> focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin.</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

Hyperbolic structures on Houghton groups

  • 期刊论文

  • 发表日期：

    2025年12月4日

查看全文
  • Anthony Genevois,
  • Geoffrey Tournier

摘要

Given a group G, its poset of hyperbolic structures \(\mathcal {H}(G)\) H ( G ) encodes all the possible cobounded actions of G on hyperbolic spaces. In this article, we describe the poset \(\mathcal {H}(H_n)\) H ( H n ) for every Houghton group \(H_n\) H n , \(n \ge 2\) n ≥ 2 . In particular, we show that \(H_n\) H n admits exactly n focal hyperbolic structures. As an application, we construct the first example of a group admitting exactly one focal hyperbolic structure, answering a question of Abbott, Balasubramanya, and Osin.