Small doubling implies small tripling for balls of large radius

We show that if $K\ge 1$ is a parameter and $S$ is a finite symmetric subset of a group containing the identity such $\mathrm{\mid }{S}^{2n}\mathrm{\mid }\le K\mathrm{\mid }{S}^n\mathrm{\mid }$ for some integer $n\ge 2K^2$, then $\mathrm{\mid }{S}^{3n}\mathrm{\mid }\le \exp (\exp (O(K^2)))\mathrm{\mid }{S}^n\mathrm{\mid }$. Such a result was previously known only under the stronger assumption that $\mathrm{\mid }{S}^{2n+1}\mathrm{\mid }\le K\mathrm{\mid }{S}^n\mathrm{\mid }$. We prove similar results for locally compact groups and vertex-transitive graphs. We indicate some results in the structure theory of vertex-transitive graphs of polynomial growth whose hypotheses can be weakened as a result.