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<p>For a surface <i>S</i> of sufficient complexity, Dehn twists act elliptically on the arc, curve, or relative arc graph of <i>S</i>. We show that composing a Dehn twist with a shift map results in a loxodromic isometry of the relative arc graph <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1036_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="54" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathcal {A}(S,p)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>p</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for any surface <i>S</i> with an isolated puncture <i>p</i> admitting a shift map. Therefore, shift maps are not type-preserving.</p>

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Shift maps are not type-preserving

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月4日

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  • Carolyn Abbott,
  • Nicholas Miller,
  • Priyam Patel

摘要

For a surface S of sufficient complexity, Dehn twists act elliptically on the arc, curve, or relative arc graph of S. We show that composing a Dehn twist with a shift map results in a loxodromic isometry of the relative arc graph \(\mathcal {A}(S,p)\) A ( S , p ) for any surface S with an isolated puncture p admitting a shift map. Therefore, shift maps are not type-preserving.