Invariant quasimorphisms and generalized mixed Bavard duality

TL;DR

This work synthesizes the Bavard duality framework by surveying the original, generalized, mixed, and generalized mixed dualities, and then proves a new generalized mixed Bavard duality in a self-contained manner. It extends scl to chains, develops invariant and $N$-quasimorphism concepts, and introduces geometric tools such as $(G,N)$-simplicial surfaces and the filling norm to bridge algebra and geometry. The paper also analyzes the space $\mathrm{W}(G,N)$ of non-extendable quasimorphisms, establishing exact sequences, dimension results, and naturality properties that connect quasimorphisms to cohomology and scl comparisons. Collectively, these results yield a unified, proof-driven perspective on how quasimorphisms control stable/ mixed scl across elements and chains, with structural insights captured by exact sequences and duality pairings.

Abstract

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.

Figures (9)

• Figure 1: the Bavard duality and three its strengthenings
• Figure 2: $2$-simplex $\sigma$
• Figure 3: $y$ to $[g_1, x_1] \cdots [g_k, x_k]$
• Figure 4: $y$ to $x_1 \cdots x_m$
• Figure 5: a deformation retract of $S \setminus C$ to $S'$

Theorems & Definitions (162)

• Theorem 1.1: Bavard duality theorem, Bavard
• Theorem 1.2: generalized Bavard duality theorem, Calegari
• Theorem 1.3: mixed Bavard duality theorem, KKMM1
• Theorem 1.4: generalized mixed Bavard duality theorem
• Definition 2.1: quasimorphisms
• Lemma 2.2: Fekete's lemma
• Lemma 2.3: homogenization of quasimorphisms
• Lemma 2.4
• proof : Proofs of Lemma \ref{['lem=homoge1']} and Lemma \ref{['lem=homoge2']}
• Corollary 2.5