Aut-invariant quasimorphisms on groups

2022/11/02 by Francesco Fournier-Facio, Fournier-Facio, Francesco, Richard D. Wade +1 · 4 citations
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2211.00800

openalex publication_date 2022/11/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a large class of groups, we exhibit an infinite-dimensional space of homogeneous quasimorphisms that are invariant under the action of the automorphism group. This class includes non-elementary hyperbolic groups, infinitely-ended finitely generated groups, some relatively hyperbolic groups, and a class of graph products of groups that includes all right-angled Artin and Coxeter groups that are not virtually abelian. This was known for F₂ by a result of Brandenbursky and Marcinkowski, but is new even for free groups of higher rank, settling a question of Miklós Abért. The case of graph products of finitely generated abelian groups settles a question of Michal Marcinkowski. As a consequence, we deduce that a variety of Aut-invariant norms on such groups are unbounded.

Cited by

• On automorphisms, quasimorphisms, and coarse automorphisms of acylindrically hyperbolic groups
• Stable commutator length on free ℚ-groups
• The infinite dimensional geometry of conjugation invariant generating sets
• JSJ splittings for all Artin groups

Related

• Aut-invariant quasimorphisms on graph products of abelian groups
• Aut-invariant quasimorphisms on free products
• Aut-invariant norms and Aut-invariant quasimorphisms on free and surface groups
• On hyperbolicity and virtual freeness of automorphism groups
• Automorphisms of graph products of groups and acylindrical hyperbolicity