Metric criteria for fixed price of countable groups

We establish general criteria for a countable group $\Gamma$ to have fixed price 1 depending on a choice of left-invariant proper metric on $\Gamma$. We apply this criterion to show that if $\Gamma_1,\Gamma_2$ are two countable groups satisfying a certain growth condition then $\Gamma_1\times \Gamma_2$ has fixed price 1. For example, $\Gamma\times \Gamma$ has fixed price 1 for any countable group $\Gamma$.