Consider the following three properties of a general group $G$:
Algebra: $G$ is abelian and torsion-free.
Analysis: $G$ is a metric space that admits a “norm”, namely, a
translation-invariant metric $d(.,.)$ satisfying: $d(1,g^n) = |n| d(1,g)$
for all $g \in G$ and integers $n$.
Geometry: $G$ admits a length function with “saturated”
subadditivity for equal arguments: $\ell(g^2) = 2 \; \ell(g)$ for all $g
While these properties may a priori seem different, in fact they turn out
to be equivalent. The nontrivial implication amounts to saying that there
does not exist a non-abelian group with a “norm”.
We will discuss motivations from analysis, probability, and geometry;
then the proof of the above equivalences; and if time permits, the
logistics of how the problem was solved, via a
that began on a
of Terence Tao.
(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil,
Pace Nielsen, Lior Silberman, and Terence Tao.)