Polymath 14: Groups with norms

Speaker: Apoorva Khare 
Affiliation: Indian Institute of Science

Abstract

Consider the following three properties of an arbitrary group $G$:

1. Algebra: $G$ is abelian and torsion-free.

2. 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 $n$.

3. Geometry: $G$ admits a length function with "saturated" subadditivity for equal arguments: $l(g^2) = 2 l(g)$ for all $g$ in $G$.

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 connections to analysis and geometry, followed by the proof of the above equivalences. We will also see the logistics of how the problem was solved, via a PolyMath project that began on a blogpost of Terence Tao.

(Joint - as D.H.J. PolyMath - with Tobias Fritz, Siddhartha Gadgil, Pace Nielsen, Lior Silberman, and Terence Tao.)