Invariant measures of Toeplitz subshifts on non-amenable groups

Let G be a countable residually finite group (for instance, ${\mathbb F}_2$ ) and let $\overleftarrow {G}$ be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every $r\geq 1$ , we construct a Toeplitz G-subshift $(X,\sigma ,G)$ , which is an almost one-to-one extension of $\overleftarrow {G}$ , having r ergodic measures $\nu _1, \ldots ,\nu _r$ such that for every $1\leq i\leq r$ , the measure-theoretic dynamical system $(X,\sigma ,G,\nu _i)$ is isomorphic to $\overleftarrow {G}$ endowed with the Haar measure. The construction we propose is general (for amenable and non-amenable residually finite groups); however, we point out the differences and obstructions that could appear when the acting group is not amenable.