Fixed subgroups in Artin groups

Oli Jones; Nicolas Vaskou

doi:10.4171/ggd/920

Fixed subgroups in Artin groups

Oli Jones
, Nicolas Vaskou

School of Mathematical & Computer Sciences

Research output: Contribution to journal › Article › peer-review

Abstract

We study fixed subgroups of automorphisms of any large-type Artin group A_Γ. We define a natural subgroup Aut_Γ(A_Γ) of Aut(A_Γ), and for every γ∈Aut_Γ(A_Γ), we find the isomorphism type of Fix(γ) and a generating set for a finite index subgroup. We show that Fix(γ) is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of Γ. Finally, we provide a natural geometric characterisation of the subgroup Aut_Γ(A_Γ), which informally is the maximal subgroup of Aut(A_Γ) leaving the Deligne complex of AΓ invariant.

Original language	English
Journal	Groups, Geometry, and Dynamics
Early online date	12 Feb 2026
DOIs	https://doi.org/10.4171/ggd/920
Publication status	E-pub ahead of print - 12 Feb 2026

Keywords

Artin groups
automorphisms
fixed subgroups

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10.4171/ggd/920

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title = "Fixed subgroups in Artin groups",

abstract = "We study fixed subgroups of automorphisms of any large-type Artin group AΓ. We define a natural subgroup AutΓ(AΓ) of Aut(AΓ), and for every γ∈AutΓ(AΓ), we find the isomorphism type of Fix(γ) and a generating set for a finite index subgroup. We show that Fix(γ) is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of Γ. Finally, we provide a natural geometric characterisation of the subgroup AutΓ(AΓ), which informally is the maximal subgroup of Aut(AΓ) leaving the Deligne complex of AΓ invariant.",

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doi = "10.4171/ggd/920",

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AU - Jones, Oli

AU - Vaskou, Nicolas

PY - 2026/2/12

Y1 - 2026/2/12

N2 - We study fixed subgroups of automorphisms of any large-type Artin group AΓ. We define a natural subgroup AutΓ(AΓ) of Aut(AΓ), and for every γ∈AutΓ(AΓ), we find the isomorphism type of Fix(γ) and a generating set for a finite index subgroup. We show that Fix(γ) is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of Γ. Finally, we provide a natural geometric characterisation of the subgroup AutΓ(AΓ), which informally is the maximal subgroup of Aut(AΓ) leaving the Deligne complex of AΓ invariant.

AB - We study fixed subgroups of automorphisms of any large-type Artin group AΓ. We define a natural subgroup AutΓ(AΓ) of Aut(AΓ), and for every γ∈AutΓ(AΓ), we find the isomorphism type of Fix(γ) and a generating set for a finite index subgroup. We show that Fix(γ) is a finitely generated Artin group, with a uniform bound on the rank in terms of the number of vertices of Γ. Finally, we provide a natural geometric characterisation of the subgroup AutΓ(AΓ), which informally is the maximal subgroup of Aut(AΓ) leaving the Deligne complex of AΓ invariant.

KW - Artin groups

KW - automorphisms

KW - fixed subgroups

U2 - 10.4171/ggd/920

DO - 10.4171/ggd/920

M3 - Article

SN - 1661-7207

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

ER -

Fixed subgroups in Artin groups

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Keywords

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