Stably Unbounded Conjugation-Invariant Norm

Updated 18 September 2025

Stably unbounded conjugation-invariant norms are norms on groups or operator algebras that display strict linear growth under stabilization for some nontrivial elements.
They serve as key indicators of large-scale algebraic and geometric complexity, particularly in groups lacking finite C-width.
Explicit constructions in infinite braid groups and operator-theoretic settings demonstrate both the utility and challenges of these norms in various mathematical contexts.

A stably unbounded conjugation invariant norm is a conjugation-invariant norm on a group (or algebra of operators) whose stabilized value, defined via iterates or limiting processes, is strictly positive for some nontrivial elements—indicating persistent linear growth even under stabilization. Such norms capture large-scale geometric, algebraic, and dynamical complexity, and their existence, characterization, or exclusion is strongly linked to properties such as finite C-width, norm equivalence, quasi-morphisms, and algebraic invariants.

1. Definition and Fundamental Properties

A conjugation-invariant norm $L: G \to \mathbb{R}$ on a group $G$ satisfies:

$L(1) = 0$
$L(g^{-1}) = L(g)$ for all $g$
$L(gh) \leq L(g) + L(h)$
$L(ghg^{-1}) = L(h)$

The stabilization of such a norm is defined by

$sL(g) = \lim_{n \to \infty} \frac{L(g^n)}{n}$

A norm is stably unbounded if there exists $g \in G$ such that $sL(g) > 0$ . This reflects uniform linear growth of the norm for powers of $g$ , even after taking averages over iterates. The defining feature is that stabilization by iteration does not lead to bounded diameter or collapse the norm.

2. Algebraic Criteria and Finite C-Width

The existence of stably unbounded conjugation-invariant norms is tightly connected to the notion of finite C-width (Bardakov et al., 2011). A group $G$ has finite C-width if, for every conjugation-invariant generating set $S$ , there exists $k$ such that $G = (S \cup S^{-1})^k$ , i.e., every element can be written as a product of at most $k$ elements from $S$ and their inverses.

Key consequences from (Bardakov et al., 2011):

Groups of finite C-width cannot support stably unbounded conjugation-invariant norms; every such norm is bounded.
Groups failing this property (such as most free products) do admit stably unbounded norms, as certain elements have length growing arbitrarily under stabilization.
Finite C-width is stable under group extensions, but not under arbitrary free products; for example, $G = G_1 * G_2$ with $G_1, G_2 \neq \mathbb{Z}_2$ does not have finite C-width.

3. Explicit Constructions and Norm Equivalence in Infinite Braid Groups

Explicit constructions of stably unbounded conjugation-invariant norms are given in (Brandenbursky et al., 2014) and (Kimura, 2016) for the commutator subgroup $[B_\infty,B_\infty]$ of the infinite braid group. The key points are:

The biinvariant word norm and (ν, p, q)-commutator length norms are defined using products of conjugates of fixed generators, measuring minimal representations.
The signature function on closures of braids provides a ν-quasimorphism, which grows linearly on powers of certain elements, ensuring stable unboundedness.
(Kimura, 2016) proves equivalence between the (ν, p, q)-commutator length and the biinvariant word norm; both detect linear growth and thus stable unboundedness.
These constructions answer the open problem of Burago–Ivanov–Polterovich: there exist perfect groups (with trivial abelianization and stably bounded commutator length) that support stably unbounded conjugation invariant norms.

4. Quasimorphisms, Partial Quasimorphisms, and Duality

Bavard’s duality generalizes to conjugation-invariant norms and subset-controlled quasimorphisms (Kawasaki, 2016). Key relationships:

Homogeneous quasimorphisms (modulo bounded defect) detect stable growth: if $sL(g) > 0$ , there exists a quasimorphism with nonzero evaluation on $g$ .
Partial quasimorphisms (defect controlled by the norm) and their homogenization process (averaging over iterates) provide necessary and sufficient conditions for undistorted elements—those for which the word norm grows linearly (Kędra, 2022).
Conjugation-invariant norms can be stably unbounded if their associated quasimorphisms or Lipschitz functions on the group detect undistorted elements.

5. Classification via Operator Theory and Singular Limits

In the operator-theoretic setting, conjugation-invariant norms are studied for families $UAU^{-1}$ where $U$ approaches a singular operator $Z$ (Falkowski et al., 24 Oct 2024). Principal findings:

For $A \in \text{End}(V)$ (finite dimension), the supremum over all $U$ near $Z$ is finite if and only if $A$ leaves $\ker(Z)$ invariant: $A(\ker(Z)) \subseteq \ker(Z)$ .
For almost all $A$ (in measure or algebraic sense), unless this alignment holds, the norm blows up: $||UAU^{-1}|| \to \infty$ as $U \to Z$ .
The dichotomy indicates that stably unbounded conjugation invariant norms are generic near singularities unless specific invariant subspace criteria are met.

6. Relationships to Asymptotic Cones and Large-Scale Geometry

The behavior of conjugation-invariant norms on asymptotic cones of groups is explored in (Karlhofer, 2022):

Asymptotic cones equipped with conjugation-invariant norms often become contractible spaces and may have strong algebraic properties (simplicity, uniform perfectness).
For the infinite symmetric group, every element in the asymptotic cone has bounded conjugation-invariant norm, precluding stably unbounded behavior in the cone, even if the ambient group lacks finite C-width.
This demonstrates that stably unbounded behavior is sensitive to limiting constructions and the large-scale structure of the group.

7. Practical and Theoretical Implications

The existence or exclusion of stably unbounded conjugation-invariant norms has direct implications for:

Geometric group theory and the classification of groups by their diameter and width properties (Kędra et al., 2018).
Rigidity and flexibility phenomena in dynamical systems and symplectic geometry (e.g., Hofer norm on Hamiltonian diffeomorphism groups (Kawasaki, 2016)).
Stability analysis in numerical linear algebra, where ill-conditioning causes generic operator families to exhibit blowup in similarity transforms unless preserved invariants exist (Falkowski et al., 24 Oct 2024).
Rationality phenomena, as in free groups where the stable conjugation-invariant word norm takes rational values for all elements (Jaspars, 2023).

Table: Obstructions and Permissions for Stably Unbounded Conjugation-Invariant Norms

Property	Stably Unbounded Norms Possible?	Example / Citation
Finite C-width	No	(Bardakov et al., 2011)
Infinite width (failure of C-width)	Yes	Free products, (Bardakov et al., 2011)
Finitely normally generated, strong boundedness	No	SL(n, ℤ), (Kędra et al., 2018)
Infinite braid groups, commutator subgroup	Yes	(1402.31911605.02306)
Operator $A$ fails $A(\ker(Z)) \subset \ker(Z)$	Yes (near $Z$ )	(Falkowski et al., 24 Oct 2024)

The structure and existence of stably unbounded conjugation-invariant norms are determined by algebraic generation and invariance properties, the analytic structure of operator families, and large-scale geometric aspects of groups. These norms play a central role in delineating the geometric and dynamical complexity across diverse areas of mathematics.