Topological models of abstract commensurators (Q6619334)

The full solenoid over a topological space \(X\) is the inverse limit of all finite covers of \(X\). In the case where \(X\) is a compact Hausdorff space admitting a locally path-connected universal cover, the authors establish a relation between the pointed homotopy equivalences of the full solenoid over \(X\) and the abstract commensurator of the fundamental group of this space and they show that 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, they 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 study is motivated by work initiated by Biswas, Nag and Sullivan on the so-called universal hyperbolic solenoid [\textit{I. Biswas} et al., Acta Math. 176, No. 2, 145--169 (1996; Zbl 0959.32026)], and by \textit{C. Odden} [Trans. Am. Math. Soc. 357, No. 5, 1829--1858 (2005; Zbl 1077.57017)] who used this to provide a topological model of the abstract commensurator of the fundamental group of a closed surface.