Stable Commutator Length (SCL)

• Stable commutator length (SCL) is a numerical invariant that measures the asymptotic efficiency of writing group elements as products of commutators through a homogenization of commutator length.
• It bridges geometric group theory, bounded cohomology, and low-dimensional topology by employing algorithmic methods and probabilistic estimates, such as n/log(n) scaling in hyperbolic groups.
• Its computability via linear programming and deep ties to quasimorphisms highlight both its theoretical importance and the computational challenges it presents.

Stable commutator length (scl) is a function measuring the asymptotic efficiency in expressing elements of the commutator subgroup in terms of products of commutators. It is a central concept at the interface of geometric group theory, low-dimensional topology, and the theory of bounded cohomology, and serves as a quantifier for the failure of commutator length to be a true norm. Recent decades have witnessed foundational progress in understanding scl, both through algorithmic methods and via deep connections with random walks, group actions on hyperbolic spaces, and probabilistic/statistical group invariants.

1. Precise Definition, Historical Context, and Foundational Properties

Given a group $G$, the commutator length $\mathrm{cl}_G(g)$ of $g \in [G,G]$ is the minimal $n$ such that $g$ is a product of $n$ commutators. The stable commutator length is then defined as

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

which homogenizes the commutator length, ensuring subadditivity and scaling under powers, and extends by linearity to the group $B_1^H(G)$ of real group 1-boundaries modulo homogeneous relations.

Key properties:

• scl is a pseudo-norm on $B_1^H(G)$.
• By Bavard duality, it is dual to the space of homogeneous quasimorphisms modulo homomorphisms, with

$$\mathrm{scl}(g) = \sup\left\{ \frac{|\phi(g)|}{2D(\phi)} : \phi \in Q(G)\right\}$$

where $D(\phi)$ is the defect.

Scl serves as a quantitative measure of "commutator depth" and provides lower bounds for the Gromov–Thurston norm on 2-dimensional homology. It encodes the existence and rigidity of non-trivial quasimorphisms and is tightly intertwined with the structure of bounded cohomology, as in Bestvina–Fujiwara's infinite dimensionality results for groups acting on hyperbolic spaces (Calegari et al., 2010).

2. Asymptotics, Randomness, and Statistical Geometry

Sharp quantitative estimates have been established for the behavior of scl on random elements in various classes of groups:

• In hyperbolic groups and groups acting nondegenerately on hyperbolic spaces, the expected scl of a random geodesic or the endpoint of a random walk of length $n$ satisfies, with high probability,

$$C_2 \frac{n}{\log n} \leq \mathrm{scl}(g) \leq C_3 \frac{n}{\log n}$$

for explicit constants $C_2, C_3 > 0$, reflecting an $n/\log n$ scaling that fundamentally refines earlier qualitative results (Calegari et al., 2010). The same order is exhibited for scl in mapping class groups and outer automorphism groups via translation length growth in respective complexes.

• In right-angled Artin and Coxeter groups (RAAGs/RACGs), scl for individual elements has a uniform spectral gap (at least $1/2$), while for integral chains the gap is non-uniform and determined by the combinatorial structure of the defining graph (Chen et al., 2020). In free and surface groups, random elements or random walks yield scl scaling as $n/\log n$ at high probability (Calegari et al., 2010).

These results link the probabilistic geometry of groups (random walks, Markov chains) with the large-scale geometry of norm balls in the associated Banach spaces—specifically, the unit ball in finite-dimensional random subspaces of the scl norm (or its dual in homogeneous quasimorphisms) exhibits uniform geometric properties and is conjectured to approximate a cross-polytope, reinforcing the statistical rigidity of scl (Calegari et al., 2010).

3. Computability, Algorithmic Structure, and Rationality Features

The computation of scl has been formalized via linear programming structures arising from the geometry of flows on polyhedra:

• In free groups and amalgamated free products, the computation is reduced to maximizing Klein functions over cones of paired unit-outflow flows, with the associated polyhedra classified in terms of their extremal rays (fully) and, more complexly, extremal points (Brantner, 2011).
• In free products of cyclic groups and their amalgamations, scl of a chain is a piecewise rational linear function of the chain and parameters (e.g., orders of the factors), and is effected via explicit polyhedral combinatorics ("scylla" algorithm) (Walker, 2013, Susse, 2013, Chen, 2016). For a fixed chain, scl varies quasirationally in the group parameters, generalizing Alden Walker's conjecture and proven via combinatorial decomposition into rectangles, triangles, and "group teeth" (Walker, 2013, Susse, 2013).
• Complexity lower bound: In $F_2$, computing scl of efficiently encoded words is NP-hard, reducible to subset sum problems (Brantner, 2011).
• Rationality phenomena: For free groups, free products of amenable groups, groups with cyclic vertex and edge stabilizers in graphs of groups, and for non-filling curves in surfaces, scl is rational and computable via LP procedures (Susse, 2013, Chen, 2019, Forester et al., 2022, Walker, 2013). For recursively presented groups, the scl spectrum is exactly the set of nonnegative right-computable numbers, but not closed under subtraction (i.e., not every difference of scl values is again an scl) (Heuer, 2019).
• Isometric embedding: The inclusion of a free group $F_S$ into its $\mathbb{Q}$-completion $F_S^\mathbb{Q}$ is isometric for scl; similarly, the inclusion of a $\pi_1$-injective subsurface defines an isometric embedding for scl and for the relative Gromov seminorm (Fournier-Facio, 18 Jul 2025, Marchand, 2023, Chen, 2016).

4. Connections to Bounded Cohomology, Quasimorphisms, and Extension Problems

Scl is characterized via the duality with homogeneous quasimorphisms and bounded cohomology from the work of Bavard and its extensions:

• Invariant and Aut-invariant quasimorphisms play a crucial role in scl, especially in the stable mixed commutator length $scl_{G,N}$. The existence of non-extendable invariant quasimorphisms signals failure of $scl_G$ and $scl_{G,N}$ to be bi-Lipschitzly equivalent (Maruyama et al., 2022, Kawasaki et al., 2022). Extension problems for these quasimorphisms correspond to exactness properties in the five-term cohomological sequence, and obstructions are calibrated by quotient spaces $W(G,N) = Q(N)^G / (H^1(N)^G + i^*Q(G))$.
• Bavard duality for invariant quasimorphisms determines $scl_{G,N}$ as

$$\mathrm{scl}_{G,N}(x) = \sup_{[\phi] \in Q(N)^G / H^1(N)^G} \frac{|\phi(x)|}{2D(\phi)}$$

(Kawasaki et al., 2022). The dimension of such quotient spaces governs the asymptotic geometry of the mixed commutator subgroup.

• Coarse group-theoretic structure: The asymptotic (coarse) geometry of large-scale scl-metrics can be explicitly tied to the dimension of $W(G,N)$ (Kawasaki et al., 2023).

5. Applications: Mapping Class Groups, Baumslag–Solitar Groups, RAAGs, and Beyond

• Mapping class groups: Scl is positive on nontrivial elements of finite-index subgroups determined by essential chiral classes in the Nielsen–Thurston decomposition, is uniformly bounded below (gap phenomenon), and is positive throughout the Torelli group and certain infinite-index subgroups (Bestvina et al., 2013). In infinite-type mapping class groups, scl is continuous on the commutator subgroup, which is clopen, and abelianizations are discrete and finitely generated under mild hypotheses (Field et al., 2021).
• Baumslag–Solitar groups: Scl is piecewise rational linear, with a universal gap of $1/12$ for nonzero values, computable via LP methods for certain classes of elements, and bounds are sharpened under acylindricity assumptions for tree actions (Clay et al., 2013).
• RAAGs and RACGs: Scl is bounded below for individual elements, but for chains, the spectral gap is determined by graph invariants (opposite path length); scl for double chains is equivalent to half the fractional stability number, and its computation is NP-hard (Chen et al., 2020).
• One-relator groups: For relators not "filling" the surface, scl is rational, and extremal surfaces exist (Forester et al., 2022). Simplicial volume and scl are linearly related (e.g., $\|G_r\| = 4\, \mathrm{scl}_S(r) - 2$), extending to probabilistic estimates for random relators (Heuer et al., 2019).
• Free $\mathbb{Q}$-groups: Every non-identity element has positive scl; the natural embedding of the free group is isometric; the space of homogeneous quasimorphisms modulo homomorphisms is infinite-dimensional. Scl rationality for these groups is conjectured and, if established, implies rationality for surface groups due to isometric embeddings of non-orientable surface groups (Fournier-Facio, 18 Jul 2025).

6. Statistical Characterizations via Random Matrices and Stable Fourier Coefficients

Recent work demonstrates that scl and closely related invariants can be recovered from the rates of exponential decay of stable Fourier coefficients in random matrix models:

• Magee–Puder theory: For a word $w$ in a free group, the minimal exponent $\beta(w, \chi)$ governing the decay of the expected trace of $w$ evaluated on random unitary matrices $U(N)$ satisfies

$$\inf_{\chi\ \mathrm{nontrivial}} \beta(w, \chi) = 2\,\mathrm{scl}(w)$$

(Puder et al., 21 Sep 2025). Analogous interpretations apply for symmetric groups $S_N$ and wreath products $G \wr S_N$ (with stable primitivity ranks and their variants as the relevant invariants).

• Stable invariants generalizing scl (e.g., stable mod-m primitivity rank $s^m(w)$ and $s^\phi(w)$ for a non-trivial character $\phi$): These are defined via infima of normalized Euler characteristics of efficient algebraic covers, are reflected in the asymptotic exponents of non-trivial stable Fourier coefficients, and constitute a powerful bridge between the combinatorics of words, algebraic extensions, and the statistics of random matrices.
• Profinite invariance: These invariants, including scl, are provably invariants of the profinite completions of the relators, encoding subtle topological and homological data in random matrix terms.
• Implications: This identification shows that scl is not only a topological or cohomological invariant but also a "statistical invariant," detectable from large-$N$ decay in random word measures; it demonstrates a profound algebra–probability–topology interface (Puder et al., 21 Sep 2025).

7. Open Problems and Research Directions

• Rationality: The rationality of scl remains open for surface groups and free $\mathbb{Q}$-groups. The isometric embedding results reduce the latter to the former for certain non-orientable surfaces (Fournier-Facio, 18 Jul 2025).
• Classification of scl spectra: For recursively presented groups the spectrum of scl matches exactly the non-negative right-computable reals but the situation for finitely presented groups is unresolved (Heuer, 2019).
• Computational complexity: Scl calculation is NP-hard in free groups for arbitrary inputs (Brantner, 2011) and in RAAGs via their connection to the fractional stability number (Chen et al., 2020).
• Extension problems: Understanding the structure and extension problems of invariant quasimorphisms, and in particular the precise measurement of non-extendability for mixed commutator length, remain active areas with implications for rigidity, cohomological invariants, and group dynamics (Kawasaki et al., 2022, Maruyama et al., 2022).
• Connections with random matrix theory: The full scope of relating topological and statistical invariants via random matrices (beyond words in free groups, to more general algebraic contexts) offers a powerful perspective with potential applications in asymptotic representation theory and ergodic theory (Puder et al., 21 Sep 2025).

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Stable commutator length is thus a rich and multifaceted invariant: computable yet nontrivially so, deeply geometric in its origins, algebraic in its applications, and now recognized as statistico-probabilistic in its far-reaching connections to the asymptotics of random group representations and matrices. Its paper weaves together bounded cohomology, spectral geometry, probabilistic group theory, and the computational complexity of group invariants.