Let G be a finitely generated malabelian group, let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A\le {{\,\textrm{Out}\,}}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>≤</mo> <mrow> <mspace width="0.166667em" /> <mtext>Out</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> be a finitely generated subgroup, and let <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\Gamma _{G,A}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mi>G</mi> <mo>,</mo> <mi>A</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> denote the preimage of A in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${{\,\textrm{Aut}\,}}(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mspace width="0.166667em" /> <mtext>Aut</mtext> <mspace width="0.166667em" /> </mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We give a general criterion for the linearity of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\Gamma _{G,A}$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="normal">Γ</mi> <mrow> <mi>G</mi> <mo>,</mo> <mi>A</mi> </mrow> </msub> </math></EquationSource> </InlineEquation> in terms of surjections from G to finite simple groups of Lie type.

Linearity criteria for automorphism groups of malabelian groups

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2026年7月6日

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Thomas Koberda,
Mark Pengitore

摘要

Let G be a finitely generated malabelian group, let $A\le {{\,\textrm{Out}\,}}(G)$ $A \leq Out (G)$ be a finitely generated subgroup, and let $\Gamma _{G,A}$ $Γ_{G, A}$ denote the preimage of A in ${{\,\textrm{Aut}\,}}(G)$ $Aut (G)$ . We give a general criterion for the linearity of $\Gamma _{G,A}$ $Γ_{G, A}$ in terms of surjections from G to finite simple groups of Lie type.