Bavard Duality in Geometric Group Theory

• Bavard duality is a fundamental theorem in geometric group theory linking stable commutator lengths to homogeneous quasimorphisms.
• It establishes a dual formulation for both absolute and mixed commutator lengths through evaluations of G-invariant quasimorphisms.
• Its applications span rigidity phenomena, symplectic topology, and computational frameworks in braid groups and group extensions.

Bavard duality is a fundamental theorem in geometric group theory and bounded cohomology that relates algebraic expressions of group elements in terms of commutators to functional-analytic objects known as quasimorphisms. The classical form expresses the stable commutator length (scl) in a group as a supremum over normalized evaluations of homogeneous quasimorphisms. The “mixed” or “relative” form—known as mixed Bavard duality—establishes an analogous correspondence in the context of a group $G$ with a normal subgroup $N$, capturing the minimal expression of elements as products of “mixed commutators.” This duality has been significantly generalized, connecting chain-level invariants, relative Gromov norms, and bounded cohomology, and has deep implications for the structure of groups, the theory of quasimorphisms, and rigidity phenomena in geometry and topology (Kawasaki et al., 2020, Kawasaki et al., 2024, Marchand, 2023).

1. Classical Bavard Duality

For a group $G$ and an element $x$ in the commutator subgroup $[G,G]$, the commutator length $\mathrm{cl}_G(x)$ is the minimal number of commutators whose product is $x$. Its stabilization,

$$\mathrm{scl}_G(x) = \lim_{n\to\infty} \frac{\mathrm{cl}_G(x^n)}{n},$$

quantifies the “asymptotic commutator cost” of $x$. A homogeneous quasimorphism $\varphi:G\to\mathbb{R}$ is a real-valued function linear on powers, with defect

$$D(\varphi) = \sup_{g, h\in G} |\varphi(gh) - \varphi(g) - \varphi(h)|.$$

Bavard’s theorem states: $$\mathrm{scl}_G(x) = \sup_{\varphi\in Q(G)\setminus\operatorname{Hom}(G,\mathbb{R})} \frac{|\varphi(x)|}{2D(\varphi)},$$ where $Q(G)$ denotes homogeneous quasimorphisms, and $\operatorname{Hom}(G,\mathbb{R})$ the homomorphisms (Kawasaki et al., 2024).

This duality bridges geometric complexity in $G$ with properties of functionals (quasimorphisms) and forms the basis for understanding stable norms, rigidity, and bounded cohomology.

2. Mixed Bavard Duality: Definitions and Main Theorem

Given $G$ and a normal subgroup $N\triangleleft G$, a $(G,N)$-commutator is any $[g,h]=ghg^{-1}h^{-1}$ with $g\in G$, $h\in N$, and the group $[G,N]$ is generated by such elements. The $(G,N)$-commutator length $\mathrm{cl}_{G,N}(x)$ of $x\in[G,N]$ is the minimal $k$ such that

$$x = [g_1,h_1]\cdots[g_k,h_k],\quad g_i\in G,\ h_i\in N,$$

and the stable version is

$$\mathrm{scl}_{G,N}(x) = \lim_{m\to\infty} \frac{\mathrm{cl}_{G,N}(x^m)}{m}.$$

A quasimorphism $f:N\to\mathbb{R}$ is $G$-invariant if $f(gxg^{-1})=f(x)$ for all $g\in G,\, x\in N$. The space $Q^h(N)^G$ collects such homogeneous $G$-invariant quasimorphisms, while $H^1(N)^G$ is the subspace of genuine homomorphisms.

Mixed Bavard Duality Theorem ([Kawasaki, Kimura, Matsushita, Mimura]; (Kawasaki et al., 2020, Kawasaki et al., 2024)): $$\mathrm{scl}_{G,N}(x) = \frac{1}{2}\sup_{f \in Q^h(N)^G \setminus H^1(N)^G} \frac{|f(x)|}{D(f)}\qquad\text{for all }x\in[G,N].$$ This recovers the classical theorem when $N=G$.

3. Geometric, Cohomological, and Chain-Level Extensions

Mixed Bavard duality has been generalized to include:

• Chain-level duality: Calegari’s framework allows replacing elements with chains, resulting in generalized duality involving $\mathrm{scl}_G(c)$ for $1$-chains $c$ and evaluations against quasimorphisms on chains (Kawasaki et al., 2024).
• Relative Gromov norm: The duality between the $\ell^1$-norm on relative homology $H_2(G,[w];\mathbb{R})$ and extremal assertions over bounded cohomology, with the Bavard duality for the relative Gromov seminorm taking the form:

$$\|\alpha\|_1 = \sup_{u\in H^2_b(X;\mathbb{R})\setminus\{0\}} \frac{\langle u,\alpha\rangle}{\|u\|_\infty}$$

for $\alpha\in H_2(X,\gamma;\mathbb{R})$ (Marchand, 2023).

• Generalized mixed Bavard duality: For chains $c$ in a relative complex $C(G,N)$, the stable mixed commutator length is

$$\mathrm{scl}_{G,N}(c) = \sup_{\varphi\in Q(N)^G\setminus\operatorname{Hom}(N,\mathbb{R})^G} \frac{|\varphi(c)|}{2D(\varphi)},$$

unifying all previous theorems (Kawasaki et al., 2024).

4. Proof Techniques and Geometric Interpretation

The proof of mixed Bavard duality follows Bavard’s original analytic-geometric approach, refined as follows (Kawasaki et al., 2020, Kawasaki et al., 2024):

• Filling norm in bar complex: For $x\in[G,N]$, define a seminorm using the infimum of $\ell^1$-norms of chains in a suitable subcomplex. The filling norm’s stabilization relates as $4\,\mathrm{scl}_{G,N}(x)$.
• Duality via Hahn–Banach theorem: The Banach dual of the chain space captures the extremal functionals, identified with $G$-invariant homogeneous quasimorphisms on $N$.
• Geometric/simplicial model: $\mathrm{cl}_{G,N}(x)$ equals the minimal genus of an orientable surface with one boundary component, labeled in $G$ such that every triangle has at least one edge in $N$ and the boundary edge represents $x$.
• Bounded cohomology viewpoint: The duality results are interpreted through exact sequences in bounded group cohomology, controlling extension and obstructions for quasimorphisms, especially encapsulated in the space $W(G,N)$ of non-extendable quasimorphisms.

5. Structural Consequences and the Space of Non-Extendable Quasimorphisms

A central structural object is

$$W(G,N) = Q(N)^G\,/\left( \operatorname{Hom}(N,\mathbb{R})^G + i^*Q(G) \right),$$

where $i^*:Q(G)\to Q(N)^G$ is the restriction. $W(G,N)$ consists of $G$-invariant homogeneous quasimorphisms on $N$ that do not extend to $G$ (Kawasaki et al., 2024).

Notably,

• If $G/N$ is amenable and $W(G,N)=0$, or if the extension $1\to N\to G\to G/N\to 1$ virtually splits, then $\mathrm{scl}_{G,N}$ and $\mathrm{scl}_G$ are bi-Lipschitzly equivalent on $[G,N]$.
• For acylindrically hyperbolic $G$ and infinite $N$, $\dim W(G,N)=2^{\aleph_0}$, indicating a rich structure of non-extendable quasimorphisms.
• If $G/N$ is finitely generated nilpotent, $W(G,N)$ is finite-dimensional and controlled by the boundedly acyclic nature of the quotient.

Extension-obstructing cohomology sequences elucidate conditions under which mixed and absolute (stable) commutator lengths are equivalent. The cohomological context is crucial for understanding the finiteness and size of $W(G,N)$ and, hence, the gap between $\mathrm{scl}_G$ and $\mathrm{scl}_{G,N}$ (Kawasaki et al., 2024).

6. Applications and Examples

Mixed Bavard duality and its generalizations have concrete computational and conceptual implications:

• Braid groups: For $G=B_n$, $N=P_n$ (pure braids), $B_n/P_n$ is finite ($S_n$), yielding approximate equivalence of mixed and absolute commutator lengths.
• Semi-direct products: For $G=N\rtimes \Gamma$ with finite or free $\Gamma$, the invariants are bi-Lipschitz equivalent.
• Hamiltonian diffeomorphism groups: For $G=\mathrm{Symp}_0(M,\omega)$, $N=\mathrm{Ham}(M,\omega)$, there are examples where the mixed and absolute invariants diverge, reflecting symplectic rigidity and linking to heavy subset theory and Entov–Polterovich’s spectral invariants (Kawasaki, 2016).
• Graphs of groups: The duality for the relative Gromov seminorm enables explicit computation of $\mathrm{scl}$ in certain amalgamated products and HNN extensions, and shows isometric embeddings of bounded cohomologies (Marchand, 2023).

Applications extend to symplectic topology, dynamics, and the geometry of infinite groups, with the duality playing an essential organizing role in connecting commutator counts, group cohomology, and the algebraic geometry of group extensions.

7. Summary Table: Bavard Duality Variants

Setting	Invariant	Dual Space
$G$ (absolute)	$\mathrm{scl}_G(x)$	$Q(G)/\operatorname{Hom}(G,\mathbb{R})$
$(G,N)$ (mixed)	$\mathrm{scl}_{G,N}(x)$	$Q(N)^G/\operatorname{Hom}(N,\mathbb{R})^G$
Chain level	$\mathrm{scl}_G(c)$	Homogeneous quasimorphisms on chains
Relative Gromov (top./alg.)	$\|\alpha\|_1$	$H^2_b(G;\mathbb{R})$ (bounded cohom.)

The Bavard duality paradigm serves as a bridge between the algebraic decomposition of elements (commutators/mixed commutators), functional analysis (quasimorphisms and their defects), and the geometry and topology of groups, with extensive ramifications for rigidity, stable norms, and bounded cohomology (Kawasaki et al., 2020, Kawasaki et al., 2024, Marchand, 2023, Kawasaki, 2016).