Realising VCD for untwisted automorphism groups of RAAGs

TL;DR

This work advances the geometric understanding of the untwisted automorphism groups of right-angled Artin groups by linking the virtual cohomological dimension to a refined spine $\widehat{K_{\Gamma}}$ constructed via an equivariant retraction that hinges on hugging/redundancy of $\, extGamma$-Whitehead partitions. It identifies graph-theoretic conditions—namely spiky and barbed—that guarantee $ extsc{vcd}(U(A_{\Gamma})) = M(L)$ and produce a cubical complex realizing this bound; it also shows that the original spine dimension $ ext{dim}(K_{\Gamma})=M(V)$ can be strictly larger, with gaps that can be arbitrarily large demonstrated by rake graphs. The paper further provides a sufficient condition ensuring $M(L)=M(V)$ and discusses special graphs (e.g., the graph $\Delta$) that illuminate the limits of such conditions. Together, these results yield both practical lower/upper bounds for $ extsc{vcd}(U(A_{\Gamma}))$ and insight into when geometric models realise the cohomological dimension, with potential algorithmic and structural implications for RAAG automorphisms.

Abstract

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 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 $U(A_Γ)$.

Figures (13)

• Figure 1: The 2-rake $T_2$.
• Figure 2: The graph $\Delta$.
• Figure 3: $v$ is a principal vertex but is not maximal.
• Figure 4: Here we have a graph $\Gamma$ and a diagram illustrating the $\Gamma$-partition $\mathcal{P} = \left(\{v, x, C_2^\pm\} \; | \; \{v^{-1}, x^{-1}, y, y^{-1}, C_1^\pm\} \; | \; \operatorname{lk}({\mathcal{P}}) \right)$, which is based at $v$. In the illustration of $\mathcal{P}$, the nodes are arranged in pairs, with a vertex of $\Gamma$ on the top row and its inverse below it on the bottom row. Here, $v$ is the only possible base for $\mathcal{P}$. The connected components of $\Gamma^\pm \setminus \operatorname{st}({v})^\pm$ are the singletons $\{x\}$, $\{x^{-1}\}$, $\{y\}$, $\{y^{-1}\}$, along with the components $C_1^\pm$, $C_2^\pm$ as indicated in the diagram.
• Figure 5: An example of a graph $\Gamma$ and three $\Gamma$-partitions, $\mathcal{P}_1$ (based at $a$), $\mathcal{P}_2$ (based at $b$), and $\mathcal{P}_3$ (based at $d$). Observe that although $e$ is split by $\mathcal{P}_2$ and $\mathcal{P}_3$, it cannot serve as a base for either as $\operatorname{lk}({e}) = \emptyset$, so $e < b$ and $e < d$. $\mathcal{P}_1$ is compatible with $\mathcal{P}_2$ since there is a choice of side of each (shaded) which have empty intersection (in fact, $\mathcal{P}_1$ and $\mathcal{P}_2$ are also adjacent, since $\operatorname{max}({\mathcal{P}_2}) = \{b\} \subseteq \{b, b^{-1}\} = \operatorname{lk}({\mathcal{P}_1})$). On the other hand, $\mathcal{P}_3$ is compatible with neither $\mathcal{P}_1$ nor $\mathcal{P}_2$. Since $d$ does not commute with $a$ or $b$, we cannot have $d$ in the link of either $\mathcal{P}_1$ or $\mathcal{P}_2$, so $\mathcal{P}_3$ is not adjacent to $\mathcal{P}_1$ or $\mathcal{P}_2$. One can verify that each side of $\mathcal{P}_3$ has non-empty intersection with each of $P_1$, $\overline{P_1}$, $P_2$, $\overline{P_2}$.

Theorems & Definitions (73)

• Theorem A: \ref{['cor:conditions 1 & 2 + barbed implies vcd = M(L)']}
• Corollary B: \ref{['cor:arbitrarily large gaps']}
• Corollary C: \ref{['cor:counterexample to suff condition hoped-theorem']}
• Theorem D: §\ref{['sec:retraction process']}, \ref{['lem:barbed implies oversize contains hugged']}
• Proposition E: \ref{['prop:M(L), M(V) for rake graphs']}
• Remark 1.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.5: cf. BrownCohomologyofGroups, Lemma VIII.2.1