Ends of a group and Spherical growth function

Question

I am trying to get some intuition about a spherical growth function of a group, using the notation from A. Mann $s_G(m)=\sum_{n=0}^m a_G(n)$, where $s_G(m)$ is the cumulative growth function and $a_G(n)$ the strict growth function.

For example for $G=\mathbb{Z}$ with one generator, $a_G(n)=2$ for all $n$. And in general, I can not assume that $a_G(n)$ is not eventually constant, for example for free groups, $a_G(n)$ increases as long as $n$ does, but as I mentioned, in $\mathbb{Z}$ it doesn't. And my intuition tells me that if $G$ has one end or infinite ends, $a_G(n)$ has enough "space" to grow together with $n$. This is something like:

CLAIM: If $G$ is a group with one end or infinite ends, then $a_G(n)$ is not eventually constant. Does this make sense? I don't have any ideas trying to prove this statement. Or there are groups that have different behaviors? Any intuition about this would be appreciated.

Edit: Maybe I need another hypothesis similar to: If $\Gamma$ is the Cayley Graph of $G$ with a finite group of generators, and it has the property that $\Gamma - B_n(u_0)$ is a connected graph where $B_n(u_0)$ is the ball with center $u_0$ and $n\in \mathbb{N}$ for any vertex $u_0$ and $n\in \mathbb{N}$??

Answer 1

Suppose $a_G(n)$ is eventually constant, and that constant is not zero. Then the cumulative growth function has linear bounds above and below.

By Gromov's polynomial growth theorem, $G$ has a finite index nilpotent subgroup.

The Bass-Guivarch theorem gives the exact degree of polynomial growth of a finitely generated nilpotent group, and in particular a nilpotent group that has a cumulative growth function with linear bounds above and below has a finite index subgroup isomorphic to $\mathbb Z$.

The group $G$ therefore has a finite index subgroup isomorphic to $\mathbb Z$, and so $G$ has two ends.