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<p>For a fixed element <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(g\in \textrm{SL}\hspace{0.55542pt}(2,\mathbb {C})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>g</mi> <mo>∈</mo> <mtext>SL</mtext> <mspace width="0.55542pt" /> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and a word <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(w=[x^n\!,y^m]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>w</mi> <mo>=</mo> <mo stretchy="false">[</mo> <msup> <mi>x</mi> <mi>n</mi> </msup> <mspace width="-0.166667em" /> <mo>,</mo> <msup> <mi>y</mi> <mi>m</mi> </msup> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation> we consider the automorphism group <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\textrm{Aut}\hspace{0.55542pt}(S_{g})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Aut</mtext> <mspace width="0.55542pt" /> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>g</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the affine threefold <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(S_{g}=\{(x,y)\in \textrm{SL}\hspace{0.55542pt}(2,\mathbb {C})^2 \,{ |}\, w(x,y)=g\}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>S</mi> <mi>g</mi> </msub> <mo>=</mo> <mrow> <mo stretchy="false">{</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mtext>SL</mtext> <mspace width="0.55542pt" /> <msup> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo>,</mo> <mi mathvariant="double-struck">C</mi> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> <mspace width="0.166667em" /> <mo stretchy="false">|</mo> <mspace width="0.166667em" /> <mi>w</mi> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi>g</mi> <mo stretchy="false">}</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We prove that the Makar-Limanov invariant <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\textrm{ML}(S_{g})=\mathscr {O}(S_{g})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>ML</mtext> <mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>g</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mi mathvariant="script">O</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>g</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\textrm{Aut}\hspace{0.55542pt}(S_{g})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>Aut</mtext> <mspace width="0.55542pt" /> <mo stretchy="false">(</mo> <msub> <mi>S</mi> <mi>g</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is Jordan.</p>

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ML-invariant and automorphism groups of certain word varieties in \(\textrm{SL}\hspace{0.55542pt}(2,C)\)

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  • 发表日期：

    2026年4月9日

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  • Tatiana Bandman

摘要

For a fixed element \(g\in \textrm{SL}\hspace{0.55542pt}(2,\mathbb {C})\) g ∈ SL ( 2 , C ) and a word \(w=[x^n\!,y^m]\) w = [ x n , y m ] we consider the automorphism group \(\textrm{Aut}\hspace{0.55542pt}(S_{g})\) Aut ( S g ) of the affine threefold \(S_{g}=\{(x,y)\in \textrm{SL}\hspace{0.55542pt}(2,\mathbb {C})^2 \,{ |}\, w(x,y)=g\}\) S g = { ( x , y ) ∈ SL ( 2 , C ) 2 | w ( x , y ) = g } . We prove that the Makar-Limanov invariant \(\textrm{ML}(S_{g})=\mathscr {O}(S_{g})\) ML ( S g ) = O ( S g ) and \(\textrm{Aut}\hspace{0.55542pt}(S_{g})\) Aut ( S g ) is Jordan.