Growth of Approximate Groups in Hyperbolic Groups

We prove a growth dichotomy for infinite approximate groups, and more generally approximate semigroups, in hyperbolic groups. If $G$ is a finitely generated hyperbolic group and $A\subseteq G$ is infinite with $$A^2\subseteq AX$$ for some finite $X\subseteq G$, then either $⟨A⟩$ is virtually cyclic, or $A$ has positive exponential growth in the ambient word metric. We also introduce a product-growth criterion for the existence of growth rates of approximate semigroups. The criterion applies to hyperbolic groups: if $G$ is hyperbolic with finite generating set $S$, then there is a constant $c_{G,S}>0$ such that $$\mathrm{\mid }UV\mathrm{\mid }\ge c_{G,S}\frac{\mathrm{\mid }U\mathrm{\mid }\mathrm{\mid }V\mathrm{\mid }}{n+k+1},U\subseteq B_n,V\subseteq B_k\mathrm{.}$$ The linear loss is optimal in order whenever $G$ contains an element of infinite order. In the free group with its standard generating set one may take $c_{G,S}=1\mathrm{/}4$. We also prove that, in a free group, if $U\subseteq S_n$ and $V\subseteq S_k$, then $$\mathrm{\mid }UV\mathrm{\mid }\ge (\frac{2}{3}+\frac{1}{3\cdot 4^{\min \{n,k\}}})\mathrm{\mid }U\mathrm{\mid }\mathrm{\mid }V\mathrm{\mid },$$ and this constant is sharp for all $n,k$.