Generators for automorphisms of special groups

TL;DR

This work establishes that Out$(G)$ is finitely generated for all compact special groups, and it provides a virtual generating set consisting of Dehn twists and pseudo-twists. Central to the method is a hierarchical shortening argument that operates along a finite Aut$(G)$-invariant hierarchy built from extended factors, standard virtual products, and salient abelian subgroups, while accounting for potential poison subgroups that obstruct twist-based generation. The paper also extends the theory to coarse-median preserving automorphisms, showing they are virtually generated by centraliser twists in a controlled, hierarchy-respecting manner. Collectively, these results generalize known finite-generation phenomena for hyperbolic and RAAG-related groups to the broader class of special groups, with broad implications for understanding automorphism groups in geometric group theory.

Abstract

Let $G$ be a (compact) special group in the sense of Haglund and Wise. We show that ${\rm Out}(G)$ is finitely generated, and provide a virtual generating set consisting of Dehn twists and ``pseudo-twists''. We exhibit instances where Dehn twists alone do not suffice and completely characterise this phenomenon: it is caused by certain abelian subgroups of $G$, called ``poison subgroups'', which can be removed by replacing $G$ with a finite-index subgroup. Similar results hold for coarse-median preserving automorphisms, without the pathologies: For every special group $G$, the coarse-median preserving subgroups ${\rm Out}(G,[μ])\leq{\rm Out}(G)$ are virtually generated by finitely many Dehn twists with respect to splittings of $G$ over centralisers. Proofs are based on a novel, hierarchical version of Rips and Sela's shortening argument.

Theorems & Definitions (171)

• Theorem 1
• Proposition 2
• Theorem 3
• Theorem 4
• Theorem 5
• Lemma 2.1
• Lemma 2.2
• proof
• Remark 2.3
• Remark 2.4