Invariant measures of Toeplitz subshifts on non-amenable groups

Let G be a countable residually finite group (for instance, F₂ ) and let G be a totally disconnected metric compactification of G equipped with the action of G by left multiplication. For every r≥ 1 , we construct a Toeplitz G -subshift (X,σ ,G) , which is an almost one-to-one extension of G , having r ergodic measures ν ₁, … ,ν ᵣ such that for every 1≤ i≤ r , the measure-theoretic dynamical system (X,σ ,G,ν ᵢ) is isomorphic to 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.