Topological models of abstract commensurators

The full solenoid over a topological space X is the inverse limit of all finite covers. When X is a compact Hausdorff space admitting a locally path-connected universal cover, we relate the pointed homotopy equivalences of the full solenoid to the abstract commensurator of the fundamental group π₁(X). The relationship is an isomorphism when X is an aspherical CW complex. If X is additionally a geodesic metric space and π₁(X) is residually finite, we show that this topological model is compatible with the realization of the abstract commensurator as a subgroup of the quasi-isometry group of π₁(X). This is a general topological analog of work of Biswas, Nag, Odden, Sullivan, and others on the universal hyperbolic solenoid, the full solenoid over a closed surface of genus at least two.