Quasimorphisms and laws

A basic reference for the background to this post is my monograph.

Let G be a group, and let [G,G] denote the commutator subgroup. Every element of [G,G] can be expressed as a product of commutators; the commutator length of an element g is the minimum number of commutators necessary, and is denoted \text{cl}(g). The stable commutator length is the growth rate of the commutator lengths of powers of an element; i.e. \text{scl}(g) = \lim_{n \to \infty} \text{cl}(g^n)/n. Recall that a group G is said to satisfy a law if there is a nontrivial word w in a free group F for which every homomorphism from F to G sends w to \text{id}.

The purpose of this post is to give a very short proof of the following proposition (modulo some background that I wanted to talk about anyway):

Proposition: Suppose G obeys a law. Then the stable commutator length vanishes identically on [G,G].

The proof depends on a duality between stable commutator length and a certain class of functions, called homogeneous quasimorphisms

Definition: A function \phi:G \to \mathbb{R} is a quasimorphism if there is some least number D(\phi)\ge 0 (called the defect) so that for any pair of elements g,h \in G there is an inequality |\phi(x) + \phi(y) - \phi(xy)| \le D(\phi). A quasimorphism is homogeneous if it satisfies \phi(g^n) = n\phi(g) for all integers n.

Note that a homogeneous quasimorphism with defect zero is a homomorphism (to \mathbb{R}). The defect satisfies the following formula:

Lemma: Let f be a homogeneous quasimorphism. Then D(\phi) = \sup_{g,h} \phi([g,h]).

A fundamental theorem, due to Bavard, is the following:

Theorem: (Bavard duality) There is an equality \text{scl}(g) = \sup_\phi \frac {\phi(g)} {2D(\phi)} where the supremum is taken over all homogeneous quasimorphisms with nonzero defect.

In particular, \text{scl} vanishes identically on [G,G] if and only if every homogeneous quasimorphism on G is a homomorphism.

One final ingredient is another geometric definition of \text{scl} in terms of Euler characteristic. Let X be a space with \pi_1(X) = G, and let \gamma:S^1 \to X be a free homotopy class representing a given conjugacy class g. If S is a compact, oriented surface without sphere or disk components, a map f:S \to X is admissible if the map on \partial S factors through \partial f:\partial S \to S^1 \to X, where the second map is \gamma. For an admissible map, define n(S) by the equality [\partial S] \to n(S) [S^1] in H_1(S^1;\mathbb{Z}) (i.e. n(S) is the degree with which \partial S wraps around \gamma). With this notation, one has the following:

Lemma: There is an equality \text{scl}(g) = \inf_S \frac {-\chi^-(S)} {2n(S)}.

Note: the function -\chi^- is the sum of -\chi over non-disk and non-sphere components of S. By hypothesis, there are none, so we could just write -\chi. However, it is worth writing -\chi^- and observing that for more general (orientable) surfaces, this function is equal to the function \rho defined in a previous post.

We now give the proof of the Proposition.

Proof. Suppose to the contrary that stable commutator length does not vanish on [G,G]. By Bavard duality, there is a homogeneous quasimorphism \phi with nonzero defect. Rescale \phi to have defect 1. Then for any \epsilon there are elements g,h with \phi([g,h]) \ge 1-\epsilon, and consequently \text{scl}([g,h]) \ge 1/2 - \epsilon/2 by Bavard duality. On the other hand, if X is a space with \pi_1(X)=G, and \gamma:S^1 \to X is a loop representing the conjugacy class of [g,h], there is a map f:S \to X from a once-punctured torus S to X whose boundary represents \gamma. The fundamental group of S is free on two generators x,y which map to the class of g,h respectively. If w is a word in x,y mapping to the identity in G, there is an essential loop \alpha in S that maps inessentially to X. There is a finite cover \widetilde{S} of S, of degree d depending on the word length of w, for which \alpha lifts to an embedded loop. This can be compressed to give a surface S' with -\chi^-(S') \le -\chi^-(\widetilde{S})-2. However, Euler characteristic is multiplicative under coverings, so -\chi^-(\widetilde{S}) = -\chi^-(S)\cdot d. On the other hand, n(S') = n(\widetilde{S})=d so \text{scl}([g,h]) \le 1/2 - 1/d. If G obeys a law, then d is fixed, but \epsilon can be made arbitrarily small. So G does not obey a law. qed.

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3 Responses to Quasimorphisms and laws

  1. Tereez says:

    Hmm, veeery interesting-reminds me of a danish I once ate. qed.

  2. Pingback: Combable functions « Geometry and the imagination

  3. Pingback: van Kampen soup and thermodynamics of DNA « Geometry and the imagination

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