Group of exponential growth always contains a free sub-group? I am not very conversant with the growth of a group, so this may be a very silly question.
It is known that $F_2$, the free group of rank $2$, has exponential growth. I was wondering whether the following is true:
If a group has exponential growth does it contain a free subgroup?
 A: Not necessarily. -- For example, the lamplighter group has exponential growth, but does not have a free subgroup of rank 2 (if $a$ and $b$ are two elements of that group of infinite order, then there are always nonzero integers $i$ and $j$ such that $a^ib^j$ is either the identity or has order $2$).
A: A famous theorem of Wolf shows that the growth of a solvable group is either polynomial or exponential. So no intermediate growth among solvable groups. And a famous theorem of Gromov shows that having polynomial growth implies being virtually nilpotent. Consequently, any solvable group that is not virtually nilpotent provides an example of a group with exponential growth but no non-abelian free subgroups.
Of course, using big theorems is not necessary to find explicit examples, but it gives some general perspectives, and it justifies that many examples exist. One simple example is the Baumslag-Solitar group $BS(1,2)$. It has a Cayley graph that is sufficiently simple to be drawn.


The pictures are taken from Wikipedia, where there is also a nice animation.
