2020/07/05 by Morimichi Kawasaki, Mitsuaki Kimura, Kawasaki, Morimichi +5 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Topology (math.AT) #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR)
paper · pdf · doi:10.48550/arxiv.2007.02257
openalex publication_date 2020/07/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let N be a normal subgroup of a group G. A quasimorphism f on N is G-invariant if f(gxg-1) = f(x) for every g ∈ G and every x ∈ N. The goal in this paper is to establish Bavard's duality theorem of G-invariant quasimorphisms, which was previously proved by Kawasaki and Kimura in the case N = [G,N]. Our duality theorem provides a connection between G-invariant quasimorphisms and (G,N)-commutator lengths. Here for x ∈ [G,N], the (G,N)-commutator length clG,N(x) of x is the minimum number n such that x is a product of n commutators which are written as [g,x] with g ∈ G and h ∈ N. In the proof, we give a geometric interpretation of (G,N)-commutator lengths. As an application of our Bavard duality, we obtain a sufficient condition on a pair (G,N) under which sclG and sclG,N are bi-Lipschitzly equivalent on [G,N].