Growth conditions for freeness of the Furstenberg boundary action

Given a finitely generated group $\mathrm{\Gamma }$ and $g\in \mathrm{\Gamma }$, we prove sufficient conditions in terms of various growth/decay functions for freeness of the action of $g$ on the Furstenberg boundary of $\mathrm{\Gamma }$. In this context, we also give a description of the support of stationary states on the reduced C*-algebra of $\mathrm{\Gamma }$.