# 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?

• It's enough to contain a free subsemigroup. Many solvable groups have free subsemigroups but none have free subgroups. There are more involved examples without free semigroups May 1 at 12:31
• Amenable groups cannot contain $F_2$, but there are amenable groups of exponential growth. May 1 at 13:55
• Many groups admit a free semigroup of rank 2 and hence have exponential growth, yet are solvable. For instance any semidirect product $\mathbf{Z}^n\rtimes_A\mathbf{Z}$ with $A$ not virtually unipotent has a free subsemigroup of rank 2.
– YCor
May 1 at 14:24

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.
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 $$2$$).
• The solvable Baumslag-Solitar groups $\langle x,y \mid y^{-1}xy=x^k \rangle$ for $|k| > 1$ are examples. May 1 at 13:52
• A semidirect product $\mathbb Z^2\rtimes \mathbb Z$ is solvable and finitely presented. If the action is hyperbolic, as is generically true, it has exponential growth. May 1 at 14:10