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 $\pi_1(X)$. The relationship is an isomorphism when $X$ is an aspherical CW complex. If $X$ is additionally a geodesic metric space and $\pi_1(X)$ 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 $\pi_1(X)$. This is a general topological analogue 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.