Fixed Subgroups of Right-Angled Artin Group Automorphisms

Definitions and mathematical statement

For a simple undirected graph $\Gamma=(V,E)$, the associated right-angled Artin group is \[ A_\Gamma=\left\langle v\ (v\in V)\ \middle|\ vw=wv\text{ whenever }\{v,w\}\in E\right\rangle. \] The supplied record does not specify whether the defining graph must be finite; that restriction remains unresolved. An automorphism $\phi\in\operatorname{Aut}(A_\Gamma)$ is a bijective group homomorphism from $A_\Gamma$ to itself. Its fixed subgroup is \[ \operatorname{Fix}(\phi)=\{g\in A_\Gamma:\phi(g)=g\}\leq A_\Gamma. \] Research question. Given a right-angled Artin group $A_\Gamma$ and an arbitrary automorphism $\phi\in\operatorname{Aut}(A_\Gamma)$, describe the subgroup $\operatorname{Fix}(\phi)$.

This preserves the supplied question, but it does not specify a particular property to establish or a proposed classification. Consequently, the record does not determine a unique conjectural assertion. No additional restriction on the automorphism is supplied.

Short English statement

Describe the subgroup of elements fixed by an arbitrary automorphism of a right-angled Artin group. The source leaves both the intended form of the description and the finiteness of the defining graph unspecified.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30004628$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The supplied record lists proposed year $2021$ and names no proposer.
• Supplied report citation: Nonpositively Curved Complexes (2022), Oberwolfach Reports, pp. 383--417, DOI 10.4171/owr/2021/8. https://doi.org/10.4171/owr/2021/8. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Jones--Vaskou, fixed subgroups in Artin groups. https://arxiv.org/abs/2407.11839.
• Supplied literature reference: Lei--Zhang, fixed subgroups. https://doi.org/10.1016/j.jalgebra.2023.07.006.
• Supplied literature reference: Oberwolfach report. https://doi.org/10.4171/owr/2021/8.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, states that fixed subgroups are understood for important subclasses while unrestricted automorphisms of general Artin or right-angled Artin groups remain unresolved, and says recent work explicitly poses the unrestricted large-type case as open. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. The record itself requests independent verification of time-sensitive status and cited claims.