In order to define stable commutator length, we first need some background:
Definition: Recall that a group \(\langle G, *\rangle\) is a set \(G\) equiped with a binary operation \(*\) such that:
- The operation is associative
- There exists an identity such that for every we have \(\mathrm{id}_G*g=g*\mathrm{id}_G=g\)
- For every \(g \in G\), there exists an inverse \(g^{-1} \in G\) such that \(g * g^{-1} =g^{-1}*g=\mathrm{id}_G\).
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 \(\{\mathrm{id}_G,a,b,a^{-1},b^{-1},ab,ab^{-1},\cdots\}\) 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 \([x, y] = \mathrm{id}_G\) 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 \(\operatorname{cl}_G(x)\) 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 \(\operatorname{cl}_G(aabbAABB) =1\)
- \(abABabAB = (abAB)^2 = [a, b] * [a, b]\), and actually \(\operatorname{cl}_G((abAB)^2) =2\)
- Although \((abAB)^3=[a,b]^3\), there is a more efficient expression \((abAB)^3 = [abA, BabAA] * [BB, ba]\), and actually \(\operatorname{cl}_G((abAB)^3) =2\) (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 \(\mathrm{scl}_G(x)\) of a word \(x \in [G,G]\) is defined as
\(\mathrm{scl}_G(x) = \lim_{n \to \infty} \frac{\mathrm{cl}_G(x^n)}{n},\)
where the limit exists since the sequence \(\mathrm{cl}_G(x^n)\) is subadditive.
Example: For a free group \(G\) generated by \(x,y\), it is known that \(\mathrm{cl}([x,y]^n)=[\frac{n}{2}]+1\) for all \(n\ge1\), so \(\mathrm{scl}_G([x, y]) = \frac{1}{2}\).
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 \(\mathrm{scl}(g)\ge\frac{1}{2}\) 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 \(\mathrm{scl}_G(g)\ge\mathrm{scl}_H(f(g))\) for any \(g\in[G,G]\), and similarly for cl.
Invariance: cl and scl are invariant under isomorphism.
Invariance under conjugation: cl and scl are invariant under conjugation. For example, \(\mathrm{scl}_G(abAB) = \mathrm{scl}_G(bABa)\).