Gromov (1981) proved that the Cayley graph of a finitely-generated group has polynomial growth if and only if the group is nilpotent. Gromov's remarkable proof shows that one can "zoom out" of the discrete graph until it looks like a continuous object. Then one uses the (difficult) solution to Hilbert's fifth problem to classify the limit objects and recover an algebraic structure on the group.

 

A few years ago, Bruce Kleiner showed that one can make this leap from geometry to algebra using spectral graph theory!  In particular, this dispenses with the need to pass to a continuous limit object. The proof becomes almost entirely "elementary." I'll explain Kleiner's proof, starting with finite groups, and then moving on to the infinite case. No significant background will be assumed. In particular, I'll explain what a nilpotent group is and give some examples (beyond the case of abelian groups).