The interaction strength I(X) of a compact hyperbolic surface X is the best upper bound for the intersection number of two closed geodesics divided by the product of their lengths. Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}_g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation> be the moduli space of compact hyperbolic surfaces of genus g and <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\text {sys}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>sys</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> the length of a shortest closed geodesic on <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$X \in \mathcal {M}_g$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>∈</mo> <msub> <mi mathvariant="script">M</mi> <mi>g</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>. We determine the asymptotic behavior of I(X), as <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="63" /> </InlineMediaObject> <EquationSource Format="TEX">$X \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}_g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation>, in terms of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq2.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\text {sys}(X)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>sys</mtext> <mo stretchy="false">(</mo> <mi>X</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We also determine the approximate behavior of the minimum of I(X) over <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathcal {M}_g$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">M</mi> <mi>g</mi> </msub> </math></EquationSource> </InlineEquation>, as <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1041_Article_IEq8.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">$g \rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>.

Intersection number, length, and systole on compact hyperbolic surfaces

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2025年9月11日

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Tina Torkaman

摘要

The interaction strength I(X) of a compact hyperbolic surface X is the best upper bound for the intersection number of two closed geodesics divided by the product of their lengths. Let $\mathcal {M}_g$ $M_{g}$ be the moduli space of compact hyperbolic surfaces of genus g and $\text {sys}(X)$ $sys (X)$ the length of a shortest closed geodesic on $X \in \mathcal {M}_g$ $X \in M_{g}$ . We determine the asymptotic behavior of I(X), as $X \rightarrow \infty $ $X \to \infty$ in $\mathcal {M}_g$ $M_{g}$ , in terms of $\text {sys}(X)$ $sys (X)$ . We also determine the approximate behavior of the minimum of I(X) over $\mathcal {M}_g$ $M_{g}$ , as $g \rightarrow \infty $ $g \to \infty$ .