Invariant measures of Toeplitz subshifts on non-amenable groups

Abstract 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, , G), which is an almost one-to-one extension of G, having r ergodic measures ₁, , ᵣ such that for every 1 i r, the measure-theoretic dynamical system (X, , G, 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.