Realising VCD for untwisted automorphism groups of RAAGs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917349052907520 |
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| author | Corrigan, Gabriel |
| author_facet | Corrigan, Gabriel |
| contents | The virtual cohomological dimension of~$\operatorname{Out}(F_n)$ 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~$A_Γ$ does not necessarily match the virtual cohomological dimension~$\textsc{vcd}(U(A_Γ))$ of the untwisted subgroup~$U(A_Γ) \leq \operatorname{Out}(A_Γ)$. Under certain graph-theoretic conditions, we perform an equivariant deformation retraction of this spine to produce a new contractible cube complex upon which~$U(A_Γ)$ acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of~$U(A_Γ)$. We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and~$\textsc{vcd}(U(A_Γ))$ can be arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_03900 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Realising VCD for untwisted automorphism groups of RAAGs Corrigan, Gabriel Group Theory Geometric Topology 20F65, 20F28, 20F36 The virtual cohomological dimension of~$\operatorname{Out}(F_n)$ 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~$A_Γ$ does not necessarily match the virtual cohomological dimension~$\textsc{vcd}(U(A_Γ))$ of the untwisted subgroup~$U(A_Γ) \leq \operatorname{Out}(A_Γ)$. Under certain graph-theoretic conditions, we perform an equivariant deformation retraction of this spine to produce a new contractible cube complex upon which~$U(A_Γ)$ acts properly and cocompactly. Furthermore, we give conditions for when the dimension of this complex realises the virtual cohomological dimension of~$U(A_Γ)$. We finish with two applications of our construction; in particular we show that the difference between the dimension of the untwisted spine and~$\textsc{vcd}(U(A_Γ))$ can be arbitrarily large. |
| title | Realising VCD for untwisted automorphism groups of RAAGs |
| topic | Group Theory Geometric Topology 20F65, 20F28, 20F36 |
| url | https://arxiv.org/abs/2501.03900 |