Aut-invariant quasimorphisms on groups

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 $F_2$ 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.

20 pages. v2: added Corollary F, Section 5.4 and Section 6.3. To appear in Transactions of the AMS

References in corpus (1)

• Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups

Cited by in corpus (2)

• The action of mapping class groups on de Rham quasimorphisms
• The space of non-extendable quasimorphisms