Metric criteria for fixed price of countable groups

Abstract

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 $\Gamma$.