Type VF for outer automorphism groups of large-type Artin groups

Given a connected large-type Artin group $A_{\mathrm{\Gamma }}$, we introduce a deformation space $\mathcal{D}$. If $\mathrm{\Gamma }$ is triangle-free, or has all labels at least 6, we show that this space is canonical, in that it depends only on the isomorphism type of $A_{\mathrm{\Gamma }}$, and admits an $\text{Out}(A_{\mathrm{\Gamma }})$-action. Using this action we conclude that $\text{Out}(A_{\mathrm{\Gamma }})$ is of type VF, which implies $\text{Out}(A_{\mathrm{\Gamma }})$ finitely presentable. We emphasise that our proof can handle cases where $\mathrm{\Gamma }$ has separating vertices, which were previously problematic. In fact, our proof works for all connected large-type Artin groups satisfying the technical condition of having rigid chunks. We conjecture that all connected large-type Artin groups have rigid chunks, and therefore outer automorphism groups of type VF.