Lengths on Free groups

Abstract:
Terence Tao posted on his blog a question of Apoorva Khare,
asking whether the free group on two generators has a length
function $l:F_2→R$ which is homogeneous, i.e., such that
$l(g^n)=n l(g)$. A week later, the problem was solved by an
active collaboration of several mathematicians (with a little help
from a computer) through Tao’s blog. In fact a more general result
was obtained, namely that any homogeneous length function on a
group $G$ factors through its abelianization $G/[G,G]$.
I will discuss the proof of this result and also the process of
discovery.

The unusual feature of the use of computers here was
that a computer generated but human readable proof was read,
understood, generalized and abstracted by mathematicians to
obtain the key lemma in an interesting mathematical result -
perhaps the first such instance.

I will also discuss conjugacy-invariant lengths on free groups and
an extension of the main result to quasi-homogeneous lengths.