Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations

TL;DR

This work defines the symmetric outer automorphism group $\Sigma Out(A_Γ)$ of a RAAG $A_Γ$ and constructs a contractible symmetric spine $K_Γ^Σ$, providing a proper cocompact action of $\Sigma Out(A_Γ)$. It generalizes the untwisted spine framework and shows that the virtual cohomological dimension of $\Sigma Out(A_Γ)$ equals the dimension of $K_Γ^Σ$, while also proving a VF-type finiteness result for outer automorphisms permuting a finite set of conjugacy classes. The approach blends Gamma-partitions, symmetric $\Gamma$-complexes, and blowups of the Salvetti complex, introducing a new norm $\lVert\cdot\rVert'$ to build an intermediate contractible complex $K_Γ^{\operatorname{min}(\mathcal{W})}$ and a deformation retract to the symmetric spine. The analysis leverages and extends the Charney–Stambaugh–Vogtmann machinery, adapting it to the symmetric setting and yielding a robust geometric model for $\Sigma Out(A_Γ)$ with computable vcd via Day–Sale–Wade’s relative outer automorphism framework. Overall, the results illuminate finiteness properties and cohomological invariants for symmetric RAAG automorphisms, with implications for understanding the broader automorphism geometry of RAAGs and related groups.

Abstract

We define the symmetric (outer) automorphism group of a right-angled Artin group and construct for it a (spine of) Outer space. This `symmetric spine' is a contractible cube complex upon which the symmetric outer automorphism group acts properly and cocompactly. One artefact of our technique is a strengthening of the proof of contractibility of the untwisted spine, mimicking the original proof that Culler--Vogtmann Outer space is contractible, which may be of independent interest. We apply our results to derive finiteness properties for certain subgroups of outer automorphisms. In particular, we prove that the subgroup consisting of those outer automorphisms which permute any given finite set of conjugacy classes of a right-angled Artin group is of type \emph{VF}, and we show that the virtual cohomological dimension of the symmetric outer automorphism group is equal to both the dimension of the symmetric spine and the rank of a free abelian subgroup.

Figures (7)

• Figure 1: $v$ is a principal vertex but is not maximal.
• Figure 2: 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 $d$). $\mathcal{P}_1$ is compatible with $\mathcal{P}_2$ since their respective positive sides (shaded) have empty intersection (in fact, $\mathcal{P}_1$ and $\mathcal{P}_2$ are also adjacent). 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 $b$, $\mathcal{P}_3$ is not adjacent to $\mathcal{P}_1$ or $\mathcal{P}_2$, and one can verify that each side of $\mathcal{P}_3$ has non-empty intersection with each of $P_1$, $\overline{P_1}$, $P_2$, $\overline{P_2}$. $\mathcal{P}_1$ is a symmetric $\Gamma$-partition, while $\mathcal{P}_2$ and $\mathcal{P}_3$ are not.
• Figure 3: Each node is a numbered item from CharneyStambaughVogtmann17. A directed edge from a node $A$ to a node $B$ encodes that $A$ is used in the proof of $B$. A circled label signifies that the norm defined in CharneyStambaughVogtmann17 is directly used at some point in the statement or proof of that item; an uncircled label corresponds to an item for which this is not the case.
• Figure 4: Dependency diagram for §\ref{['sec:contractibility of Kmin']} of this document. Each node either contains a result from this paper (in italics) or a result from CharneyStambaughVogtmann17. All nodes containing a result from the present document also indicate the comparable (indicated by 'cf.') or equivalent (indicated by '=') result from CharneyStambaughVogtmann17 (upright text). A directed edge from a node $A$ to a node $B$ indicates that $A$ implies, or is used in the proof of, $B$.
• Figure 5: A schematic of the decomposition of the vertices of a $\Gamma$-complex $X$ into two sets $\beta(H)$ and $\overline{\beta}(H)$, one either side of a hyperplane $H$ in a treelike set $\mathcal{T}$. Here $H_1,\; \dots,\; H_m$ are the companion hyperplanes to $H$, while the edges labelled $e_{H_i}$ are the companion edges to those labelled $e_H$. Note that $m \geq 1$ since $\Gamma$-complexes have no separating hyperplanes. $H$ is symmetric relative to $\mathcal{T}$ if and only if $m = 1$.

Theorems & Definitions (87)

• Theorem A: \ref{['thm:symspine is a retract of Kmin']}
• Theorem B: \ref{['thm:vcdsymout = dim(symspine)']}
• Theorem C: \ref{['thm:Kmin is contractible']}
• Corollary D: \ref{['cor:outw is type VF']}
• Proposition E: \ref{['prop:symout contains free abelian subgp of rank MSigma(L)']} & \ref{['prop:MSigma(L) = MSigma(V)']}
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4: cf. BrownCohomologyofGroups, Lemma VIII.2.1
• Theorem 2.5: Serre