Are free products of exponential growth?

Question

Let $G$ and $H$ be finitely generated groups. Is the free product $G*H$ always of exponential growth?

I am guessing that the answer is yes, but I don't know how to prove it correctly. My idea is to find a free subgroup or a free quotient (ping pong lemma or find some tree to act on). Another idea was to use the fact that free products have infinitely many ends and "hence" must be of exponential growth...

Edit: I now realize that the answer is no: Take the infinite dihedral group $D_\infty \cong Z_2 *Z_2$. It is easily seen that it quasi-isometric to $Z$. Hence it doesn't have exponential growth.
...So is this ever true: what conditions must I impose on $G$ and $H$?

Answer 1

I think the answer is yes unless one of the groups is trivial or both have order 2.

Suppose $1\neq g\in G$ and $1\neq h_1,h_2\in H$ with $h_1\neq h_2$. Then for any $n$, the $2^n$ (exponential in $n$) elements $gh_{i_1}gh_{i_2}\dots gh_{i_n}$ of $G\ast H$ are all different.