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<p>We show that an accessible group with infinitely many ends has property <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(R_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>. That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(R_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation> is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(R_{\infty }\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>R</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>.</p>

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Property \(R_{\infty }\) for groups with infinitely many ends

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月19日

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  • Francesco Fournier-Facio,
  • Harry Iveson,
  • Armando Martino,
  • Wagner Sgobbi,
  • Peter Wong

摘要

We show that an accessible group with infinitely many ends has property \(R_{\infty }\) R ∞ . That is, it has infinitely many twisted conjugacy classes for any twisting automorphism. We deduce that having property \(R_{\infty }\) R ∞ is undecidable amongst finitely presented groups. We also show that the same is true for a wide class of relatively hyperbolic groups, filling in some of the gaps in the literature. Specifically, we show that a non-elementary, finitely presented relatively hyperbolic group with finitely generated peripheral subgroups which are not themselves relatively hyperbolic, has property \(R_{\infty }\) R ∞ .