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<p>Given a monomorphism <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\Psi:\cal{H}\rightarrow \cal{F}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi mathvariant="normal">Ψ</mi> <mo>:</mo> <mrow> <mi mathvariant="script">H</mi> </mrow> <mo stretchy="false">→</mo> <mrow> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\cal{H}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">H</mi> </mrow> </math></EquationSource> </InlineEquation> is a proper free factor of the free group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\cal{F}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">F</mi> </mrow> </math></EquationSource> </InlineEquation>, we show the associated partial mapping torus <i>X</i> of Ψ has negative immersions iff <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\cal{H}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="script">H</mi> </mrow> </math></EquationSource> </InlineEquation> has finite height in <i>π</i><sub>1</sub><i>X</i> if and only if Ψ is fully irreducible. We survey related properties and discuss possible directions to pursue further.</p>

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Negative immersions and finite height mappings

  • 期刊论文

  • 发表日期：

    2026年2月19日

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  • Brahim Abdenbi,
  • Daniel T. Wise

摘要

Given a monomorphism \(\Psi:\cal{H}\rightarrow \cal{F}\) Ψ : H → F where \(\cal{H}\) H is a proper free factor of the free group \(\cal{F}\) F , we show the associated partial mapping torus X of Ψ has negative immersions iff \(\cal{H}\) H has finite height in π1X if and only if Ψ is fully irreducible. We survey related properties and discuss possible directions to pursue further.