  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.
