<p>The virtual cohomological dimension of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {Out}(F_n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>Out</mo> <mo stretchy="false">(</mo> <msub> <mi>F</mi> <mi>n</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> is given precisely by the dimension of the spine of Culler–Vogtmann Outer space. However, the dimension of the spine of untwisted Outer space for a general right-angled Artin group <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A_\Gamma $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> </math></EquationSource> </InlineEquation> does not necessarily match the virtual cohomological dimension <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textsc {vcd}(U(A_{\Gamma }))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mstyle mathsize="0.6em"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">D</mi> </mstyle> <mo stretchy="false">(</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> of the untwisted subgroup <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$U(A_\Gamma ) \le \operatorname {Out}(A_\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mo>Out</mo> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Under certain graph-theoretic conditions, we perform an equivariant deformation retraction of this spine to produce a new contractible cube complex upon which <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$U(A_\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$U(A_\Gamma )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\textsc {vcd}(U(A_{\Gamma }))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mstyle mathsize="0.6em"> <mi mathvariant="normal">V</mi> <mi mathvariant="normal">C</mi> <mi mathvariant="normal">D</mi> </mstyle> <mo stretchy="false">(</mo> <mi>U</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>A</mi> <mi mathvariant="normal">Γ</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> can be arbitrarily large.</p>

Realising VCD for untwisted automorphism groups of RAAGs

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2026年3月19日

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Gabriel Corrigan

摘要

The virtual cohomological dimension of $\operatorname {Out}(F_n)$ $Out (F_{n})$ is given precisely by the dimension of the spine of Culler–Vogtmann Outer space. However, the dimension of the spine of untwisted Outer space for a general right-angled Artin group $A_\Gamma $ $A_{Γ}$ does not necessarily match the virtual cohomological dimension $\textsc {vcd}(U(A_{\Gamma }))$ $V C D (U (A_{Γ}))$ of the untwisted subgroup $U(A_\Gamma ) \le \operatorname {Out}(A_\Gamma )$ $U (A_{Γ}) \leq Out (A_{Γ})$ . Under certain graph-theoretic conditions, we perform an equivariant deformation retraction of this spine to produce a new contractible cube complex upon which $U(A_\Gamma )$ $U (A_{Γ})$ acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of $U(A_\Gamma )$ $U (A_{Γ})$ . We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and $\textsc {vcd}(U(A_{\Gamma }))$ $V C D (U (A_{Γ}))$ can be arbitrarily large.