Nielsen realisation for untwisted automorphisms of right-angled Artin groups

TL;DR

This paper proves Nielsen realisation for finite subgroups of untwisted outer automorphisms of right-angled Artin groups, constructing non-positively curved cube complexes that realize these automorphism actions geometrically.

Contribution

It establishes Nielsen realisation for untwisted automorphisms of RAAGs, linking algebraic automorphisms to geometric actions on cube complexes.

Findings

Finite subgroups of untwisted automorphisms can be realized geometrically.

Constructs non-positively curved cube complexes with prescribed automorphism actions.

Bridges algebraic automorphisms and geometric group actions in RAAGs.

Abstract

We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $\Gamma$, and any finite group $G\leqslant \mathrm{U}^0(A_\Gamma) \leqslant \mathrm{Out}^0(A_\Gamma)$, we find a non-positively curved cube complex with fundamental group $A_\Gamma$ on which $G$ acts by isometries, realising the action on $A_\Gamma$.