A basic reference for the background to this post is my monograph.
Let be a group, and let
denote the commutator subgroup. Every element of
can be expressed as a product of commutators; the commutator length of an element
is the minimum number of commutators necessary, and is denoted
. The stable commutator length is the growth rate of the commutator lengths of powers of an element; i.e.
. Recall that a group
is said to satisfy a law if there is a nontrivial word
in a free group
for which every homomorphism from
to
sends
to
.
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 obeys a law. Then the stable commutator length vanishes identically on
.
The proof depends on a duality between stable commutator length and a certain class of functions, called homogeneous quasimorphisms.
Definition: A function is a quasimorphism if there is some least number
(called the defect) so that for any pair of elements
there is an inequality
. A quasimorphism is homogeneous if it satisfies
for all integers
.
Note that a homogeneous quasimorphism with defect zero is a homomorphism (to ). The defect satisfies the following formula:
Lemma: Let be a homogeneous quasimorphism. Then
.
A fundamental theorem, due to Bavard, is the following:
Theorem: (Bavard duality) There is an equality where the supremum is taken over all homogeneous quasimorphisms with nonzero defect.
In particular, vanishes identically on
if and only if every homogeneous quasimorphism on
is a homomorphism.
One final ingredient is another geometric definition of in terms of Euler characteristic. Let
be a space with
, and let
be a free homotopy class representing a given conjugacy class
. If
is a compact, oriented surface without sphere or disk components, a map
is admissible if the map on
factors through
, where the second map is
. For an admissible map, define
by the equality
in
(i.e.
is the degree with which
wraps around
). With this notation, one has the following:
Lemma: There is an equality .
Note: the function is the sum of
over non-disk and non-sphere components of
. By hypothesis, there are none, so we could just write
. However, it is worth writing
and observing that for more general (orientable) surfaces, this function is equal to the function
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 . By Bavard duality, there is a homogeneous quasimorphism
with nonzero defect. Rescale
to have defect
. Then for any
there are elements
with
, and consequently
by Bavard duality. On the other hand, if
is a space with
, and
is a loop representing the conjugacy class of
, there is a map
from a once-punctured torus
to
whose boundary represents
. The fundamental group of
is free on two generators
which map to the class of
respectively. If
is a word in
mapping to the identity in
, there is an essential loop
in
that maps inessentially to
. There is a finite cover
of
, of degree
depending on the word length of
, for which
lifts to an embedded loop. This can be compressed to give a surface
with
. However, Euler characteristic is multiplicative under coverings, so
. On the other hand,
so
. If
obeys a law, then
is fixed, but
can be made arbitrarily small. So
does not obey a law. qed.
Hmm, veeery interesting-reminds me of a danish I once ate. qed.
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