Metric criteria for fixed price of countable groups

We establish general criteria for a countable group $\mathrm{\Gamma }$ to have fixed price 1 depending on a choice of left-invariant proper metric on $\mathrm{\Gamma }$. We apply this criterion to show that if ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$ are two countable groups satisfying a certain growth condition then ${\mathrm{\Gamma }}_1\times {\mathrm{\Gamma }}_2$ has fixed price 1. For example, $\mathrm{\Gamma }\times \mathrm{\Gamma }$ has fixed price 1 for any countable group $\mathrm{\Gamma }$.