Danny Calegari

Title: stable commutator length in free groups

Abstract: if $G$ is a group, and $g$ is in $[G,G]$, the
commutator length of $g$ (denoted $\text{cl}(g))$ is the
least number of commutators in $G$ whose product is $g$,
and the stable commutator length is the limit
$\text{scl}(g) = \lim \text{cl}(g^n)/n$. Scl extends naturally
to a (pseudo) norm on $B_1(G)$, the space of (real)
group 1-boundaries. Dual to this space is the
vector space $Q(G)$ of (homogeneous) quasimorphisms on
$G$; i.e. functions $\phi:G \to \R$ satisfying
$\phi(g^n) = n \phi(g)$ for which there is a least
real number $D(\phi)$ such that $|\phi(g) + \phi(h) - \phi(gh)|$
is at most $D(\phi)$ for all $g,h$ in $G$.

The two main theorems I would like to talk about are:

Theorem 1 (Rationality): Let $F$ be a free group. Then the scl
pseudo-norm is piecewise rational linear on $B_1(F)$.

Theorem 2 (Rigidity): For every surface $S$ with $\pi_1(S) = F$
the chain $\partial S$ in $B_1(F)$ is (projectively) in the
interior of a top dimensional face of the scl norm. Moreover,
the unique homogeneous quasimorphism dual to this face (up to
$H^1$ and scale) is the rotation quasimorphism associated to
a hyperbolic structure on $S$.

I will also (if time permits) discuss scl and surgery on free
products of abelian groups, and its relation to certain
(experimentally observed) power laws in the distribution
of values of scl in free groups, and to exotic isometries
of the scl norm.