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

TL;DR

This paper introduces a deformation space for large-type Artin groups, proving that their outer automorphism groups are of type VF, which implies they are finitely presentable, even in complex cases.

Contribution

The paper develops a canonical deformation space for large-type Artin groups and proves their outer automorphism groups are of type VF, including cases with separating vertices.

Findings

Outer automorphism groups are of type VF for large-type Artin groups

The deformation space depends only on the isomorphism type of the group

Proof handles cases with separating vertices and introduces the concept of rigid chunks

Abstract

Given a connected large-type Artin group $A_\Gamma$, we introduce a deformation space $\mathcal{D}$. If $\Gamma$ 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 $A_\Gamma$, and admits an $\Out(A_\Gamma)$-action. Using this action we conclude that $\Out(A_\Gamma)$ is of type VF, which implies $\Out(A_\Gamma)$ finitely presentable. We emphasise that our proof can handle cases where $\Gamma$ 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.