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<p>In this paper, we prove the lack of asymptotic transitivity of the outer automorphism group action of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_998_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {Z}}^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> on <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_998_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(\textrm{SL}_n({\mathbb {F}}_q)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-character varieties of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_998_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\({\mathbb {Z}}^r\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mi>r</mi> </msup> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_998_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(n=2,3\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_998_Article_IEq5.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(r\ge 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Along the way, we stratify the character varieties and compute the <i>E</i>-polynomial, also known as the Hodge–Deligne polynomial or Serre polynomial, of these character varieties.</p>

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Asymptotic dynamics on character varieties over finite fields

  • 期刊论文

  • 发表日期：

    2025年3月18日

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  • Cigole Thomas

摘要

In this paper, we prove the lack of asymptotic transitivity of the outer automorphism group action of \({\mathbb {Z}}^r\) Z r on \(\textrm{SL}_n({\mathbb {F}}_q)\) SL n ( F q ) -character varieties of \({\mathbb {Z}}^r\) Z r for \(n=2,3\) n = 2 , 3 and \(r\ge 2\) r ≥ 2 . Along the way, we stratify the character varieties and compute the E-polynomial, also known as the Hodge–Deligne polynomial or Serre polynomial, of these character varieties.