Are there any usages for growth rate that are relatively easy to show?

Question

I'm giving a lecture about growth rates in a seminar. the chapter in the book I'm working with mainly focuses on computing growth rates and states Gromov's polynomial growth theorem (and the fact that it's way beyond the scope of the chapter). I feel like the chapter, and therefore my lecture, lacks concrete results that can be implied from certain growth rates. is there anything trivial or at least intuitively understandable of this sort? most of what I see on the Wikipedia page is calculations of growth rates.

Answer 1

If you already know something about growth rates from that Wikipedia page, in particular the Bass-Guivarch formula for the polynomial growth degree of a finitely generated nilpotent group, then there are a lot of reasonably quick applications of Gromov's polynomial growth theorem. Here are some of them.

Let $G$ be a finitely generated group whose growth is polynomial of degree $d \ge 1$ (if $G$ has polynomial growth degree $d=0$ then $G$ is finite; you don't need Gromov's Theorem for that).

By applying Gromov's Theorem, it follows that $G$ has a nilpotent subgroup $N$ of finite index. And then, by applying the Bass-Guivarch formula to $N$, you can make the following additional implications for low values of $d$:

1. If $d=1$ then $G$ has a finite index subgroup isomorphic to $\mathbb Z$.
2. If $d=2$ then $G$ has a finite index subgroup isomorphic to $\mathbb Z^2$.
3. If $d=3$ then $G$ has a finite index subgroup isomorphic to $\mathbb Z^3$.
4. If $d=4$ then $G$ has a finite index subgroup isomorphic to either $\mathbb Z^4$ or to a finite index subgroup of the integer Heisenberg group $$H_3(Z) = \left\{ \begin{pmatrix} 1 & K & M \\ 0 & 1 & L \\ 0 & 0 & 1 \end{pmatrix} \, : \, K, L, M \in \mathbb Z \right\}. $$

You can see that this begins to get complicated when $n=4$, and although one can continue onward it does get more and more complicated as $n$ increases.

In each case, one makes these calculations by using finite index of $N < G$ to conclude that $N$ and $G$ have the same polynomial growth degree $d$, then writing out the lower central series $$N = N_1 \, \triangleright \, N_2 \,\, \triangleright \, ... \, \triangleright \, N_K = \{\text{Id}\}, \quad \quad N_{k+1} = [N,N_k] $$ and then applying the Bass-Guivarch formula \begin{align*} d &= \sum_{k \ge 1} k \cdot \text{rank}(N_k / N_{k+1}) \\ &= 1 \cdot \text{rank}(N_1 / N_2) + 2 \cdot \text{rank}(N_2 / N_3) + 3 \cdot \text{rank}(N_3 / N_4) + \cdots \end{align*} where $\text{rank}(N_k / N_{k+1})$ denotes the rank of the free abelian summand of the finitely generated abelian group $N_k / N_{k+1}$. In the course of the calculations one will need to work with elements of the theory of lower central series, finitely generated abelian groups, and finitely generated nilpotent groups, to make the final conclusions.

For a few details in the simplest case $d=1$, by applying the Bass-Guivarch theorem one concludes that all higher commutator subgroups $N_k$ for $k \ge 2$ are finite (because $\text{rank}(N_k/N_{k+1})=0$ for $k \ge 2$), and that $N/[N,N]$ is a finitely generated abelian group of rank $1$ (because $\text{rank}(N_1/N_2) = \text{rank}(N/[N,N])=1$). It quickly follows (from the classification of finitely generated abelian groups) that $N/[N,N]$ has an infinite cyclic subgroup of finite index, hence so does $N$ and finally so does $G$.