In order to define stable commutator length, we first need some background:
Definition: Recall that a group \(\langle G, *\rangle\) is a set \(G\) equipped with a binary operation \(*\) such that:
- The operation is associative
- There exists an identity \(\text{id}_G\) such that for every \(g \in G\) we have \(\text{id}_G * g = g * \text{id}_G = g\)
- For every \(g \in G\), there exists an inverse \(g^{-1} \in G\) such that \(g * g^{-1} = g^{-1} * g = \text{id}_G\).
Definition: A free group is the largest possible group “generated” by some set. For example, let \(S = \{a,b\}\). Then the free group \(F_S\) consists of “reduced words” (i.e. \(a\) and \(a^{-1}\) cannot be adjacent and similarly for \(b, b^{-1}\)) in the letters \(a, b, a^{-1}, b^{-1}\) such as \({\text{id}_G, a, b, a^{-1}, b^{-1}, ab, ab^{-1}, \cdots}\) and so on.
Definition: For a group \(G\) with elements \(x, y in G\), define the commutator \([x, y]\) of \(x\) and \(y\) to be \([x, y] := xyx^{-1}y^{-1}\).
Note that \([x, y] = \text{id}_G\) iff \(xy = yx\). Let \([G, G]\) be commutator subgroup generated by all commutators of \(G\). Commutators give a measure of how abelian a group is, since \(G/[G, G]\) is the largest abelian quotient of \(G\).
Definition: The commutator length \(\text{cl}_G (x)\) of a word \(x \in [G, G]\) is the minimal number of commutators that can be concatenated to prepresent the given word \(x\).
Example: Let \(G = F_2\).
- \(aabbAABB = [aa, bb]\) implies \(\text{cl}_G (aabbAABB) = 1\)
- \(abABabAB = (abAB)^2 = [a, b] * [a, b]\), and actually \(\text{cl}_G ((abAB)^2) = 2\)
- Although \((abAB)^3 = [a, b]^3\), there is a more efficient expression \((abAB)^3 = [abA, BabAA] * [BB, ba]\), and actually \(\text{cl}_G ((abAB)^3) = 2\) (Culler’s identity)
Given this definition, the stable commutator length can be defined as follows, measuring the growth of the commutator length for powers of \(x\).
Definition: The stable commutator length \(\text{scl}_G (x)\) of a word \(x \in [G, G]\) is defined as
\[\text{scl}_G (x) = \lim_{n \to \infty} \frac{\text{cl}_G (x^n)}{n},\]
where the limit exists since the sequence \(\text{cl}_G (x^n)\) is subadditive.
Example: For a free group \(G\) generated by \(x, y\), it is known that \(\text{cl}([x, y]^n) = \left[\frac{n}{2}\right] + 1\) for all \(n \geq 1\), so \(\text{scl}_G([x, y]) = \frac{1}{2}\).
This holds more generally for any two non-commuting elements \(x, y\) in a free group, which follows from a “spectral gap theorem” of Duncan-Howie, which shows that \(\text{scl}(g) \geq \frac{1}{2}\) for any nontrivial element \(g\) in the free group.
Nice Properties
Monotonicity: For any homomorphism \(f : G \to H\) (i.e. a map respecting the group operations), we have \(\text{scl}_G (g) \geq \text{scl}_H (f(g))\) for any \(g \in [G, G]\), and similarly for \(\text{cl}\).
Invariance: \(\text{cl}\) and \(\text{scl}\) are invariant under isomorphism.
Invariance under conjugation: \(\text{cl}\) and \(\text{scl}\) are invariant under conjugation. For example, \(\text{scl}_G (abAB) = \text{scl}_G (bABa)\).