Gromov's Seminal Breakthrough (short)
April 21, 2017
I’m currently reading the long overdue 1981 paper of Gromov showing that virtually nilpotent groups are precisely the groups with polynomial growth.
To my surprise, the breakthrough wasn’t mind-blowingly, spectacularly, unspeakably awesomely seminalDon’t get me wrong, it’s still very seminal, just not as much as I originally thought, which would have been hard to satisfy for any mathematical result. as I thought it was, in the sense that the literature was somewhat building up to itOf course, this is a fundamental theme in mathematical research, which is easy to forget because it is routinely contradicted by myths of lone mathematicians. As you can see, I’m completely guilty of forgetting myself! .
Consider the following theorem which was generated a well-known fact by the time Gromov wrote his paper:
Theorem: The finitely generated Abelian groups have polynomial growth.This is given by the theory of Erhart polynomials.
In 1968, thirteen years before Gromov’s theorem, this result was extended to virtually nilpotent groups. Nilpotent groups are almost abelian, in some sense. The interesting part is the virtually part of the statement, meaning that polynomial growth holds even if the nilpotent part of a group is just a subgroup of finite index.
Theorem: If a finitely generated group $G$ has a nilpotent subgroup of finite index, then $G$ has polynomial growth.J. WOLF, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, J. Dzff. Geom., 2 (I968), 421-446.
This is already one direction of the theorem. Also from 1968, we have
Theorem (Milnor-Wolf): A finitely generated solvable group $G$ has exponential growth unless $G$ contains a nilpotent subgroup of finite index.
This tells us that nilpotent of finite index is a very strong property in terms of growth, with the power to turn a group that would have otherwise been of exponentialHere, exponential just means something faster than any polynomial. If $\beta$ is the growth function, then if for some real constant $C > 1$, we have $\beta(n) \geq C^n, \quad n \geq 1$, then $\beta(n)$ is exponential. growth into something with a slower growth rate.
This implies, with a 1972 theorem of TitsJ. TITS,Free subgroups in linear groups, J. Algebra, 20 (I972), 25o-27o. ,
Theorem (Tits): A finitely generated subgroup $G$ of a connected Lie group has exponential growth unless $G$ contains a nilpotent group of finite index.
Nine years later in 1981, Gromov proves that polynomiality is inherently a nilpotent property.
Theorem (Gromov): If a finitely generated group $G$ has polynomial growth, then $G$ contains a nilpotent subgroup of finite index.
Which is still impressive (soon, I will learn how he did this), but not as crazily pulled out of context as I originally thought.
Special thanks to Henry Liu for reading a draft of this article and giving insightful feedback.