Topological Models of Abstract Commensurators

TL;DR

This paper explores the relationship between the topological structure of the full solenoid over a space and the algebraic properties of its fundamental group, establishing isomorphisms and compatibilities in specific geometric and algebraic contexts.

Contribution

It establishes a topological model linking the homotopy equivalences of the full solenoid to the abstract commensurator of the fundamental group, generalizing previous work on hyperbolic surfaces.

Findings

Isomorphism between homotopy equivalences of the solenoid and the commensurator for aspherical CW complexes.

Compatibility of the topological model with the quasi-isometry group when the space is a geodesic metric space with residually finite fundamental group.

Generalizes known results from hyperbolic surface theory to broader classes of spaces.

Abstract

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.