Invariant quasimorphisms and generalized mixed Bavard duality

This article provides an expository account of the celebrated duality theorem of Bavard and three its strengthenings. The Bavard duality theorem connects scl (stable commutator length) and quasimorphisms on a group. Calegari extended the framework from a group element to a chain on the group, and established the generalized Bavard duality. Kawasaki, Kimura, Matsushita and Mimura studied the setting of a pair of a group and its normal subgroup, and obtained the mixed Bavard duality. The first half of the present article is devoted to an introduction to these three Bavard dualities. In the latter half, we present a new strengthening, the generalized mixed Bavard duality, and provide a self-contained proof of it. This third strengthening recovers all of the Bavard dualities treated in the first half; thus, we supply complete proofs of these four Bavard dualities in a unified manner. In addition, we state several results on the space $\mathrm{W}(G,N)$ of non-extendable quasimorphisms, which is related to the comparison problem between scl and mixed scl via the mixed Bavard duality.