Colloquium abstracts

Siddhartha Gadgil
Indian Institute of Science, Banagalore
January 16, 2020

Lengths on Free groups:  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.