<p>For a compact 2-orbifold with negative Euler characteristic <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathcal {O}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">O</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>, the variety of characters of <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq2.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi _1({\mathcal {O}}^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mrow> <mi mathvariant="script">O</mi> </mrow> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\text {SL}_{n}({\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is a non-singular manifold at <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>-irreducible representations. In this paper we prove that when a <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">${\mathbb {C}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">C</mi> </math></EquationSource> </InlineEquation>-irreducible representation of <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq6.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="50" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi _1(\mathcal O^2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mrow> <mo stretchy="false">(</mo> <msup> <mi mathvariant="script">O</mi> <mn>2</mn> </msup> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$\text {SL}_{n}({\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is viewed in <InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10711_2025_1011_Article_IEq8.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="66" /> </InlineMediaObject> <EquationSource Format="TEX">$\text {SL}_{n+1}({\mathbb {R}})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mtext>SL</mtext> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, then the variety of characters is singular, and we describe the singularity.</p>

Deforming reducible representations of surface and 2-orbifold groups

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2025年5月19日

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Joan Porti

摘要

For a compact 2-orbifold with negative Euler characteristic ${\mathcal {O}}^2$ $O^{2}$ , the variety of characters of $\pi _1({\mathcal {O}}^2)$ $π_{1} (O^{2})$ in $\text {SL}_{n}({\mathbb {R}})$ ${SL}_{n} (R)$ is a non-singular manifold at ${\mathbb {C}}$ $C$ -irreducible representations. In this paper we prove that when a ${\mathbb {C}}$ $C$ -irreducible representation of $\pi _1(\mathcal O^2)$ $π_{1} (O^{2})$ in $\text {SL}_{n}({\mathbb {R}})$ ${SL}_{n} (R)$ is viewed in $\text {SL}_{n+1}({\mathbb {R}})$ ${SL}_{n + 1} (R)$ , then the variety of characters is singular, and we describe the singularity.