On the minimal exponential growth rates in free products of groups

In this talk we will discuss lower bounds for minimal exponential growth rate $\Omega(G*H)$ of the free product of groups $G$ and $H$. A.Mann has proved that $\Omega(G*H)\geqslant \sqrt{2}$ for all free products $G*H$ except when it is a free product $C_2*C_2$ of two cyclic groups of order $2$. This lower bound is precise in the case of $C_2*C_3$, i.e. $\Omega(C_2*C_3)=\sqrt{2}$.
We prove that in the cases when $G*H$ is neither $C_2*C_2$ nor $C_2*C_3$, the lower bound of A.Mann can be strengthened, namely $\Omega(G*H)\geqslant \frac{1+\sqrt{5}}2$. This talk is based on a joint work with M.Bucher-Karlsson.