Amenable exponential growth

any solvable group which is not virtually nilpotent has exponential growth. For an example, take a semi-direct product of $Z$ and the direct sum $A$ of infinitely many copies of ${\mathbb Z}$ where the cyclic group acts by translations. It is not hard to see that the growth of the balls is exponential. In fact consider all sequences $(a_i)$ in $A$ with support $[1,\sqrt{n}]$ where $a_i$ is arbitrarily chosen from the set $0,1,2$. It is easy to see that the length of this element in the group is at most $$ C \sum_{i=1}^{\sqrt{n}} i a_i \le 2C \sqrt{n}^2/2= C n $$ which gives you the exponential grwoth

@ HW: One could add a polycyclic example, e.g. the semi-direct product $\mathbb{Z}^2\rtimes \mathbb{Z}$, where $\mathbb{Z}$ acts by powers of some hyperbolic automorphism, e.g. the Anosov matrix (2,1,1,1).

All solvable groups are amenable, but many have exponential growth. Look at J. Milnor's classic 1968 paper...