Compact aspherical solenoids

Abstract

We consider compact, aspherical solenoids obtained as the inverse limit of a system of CW~complexes and covering maps. This includes $P$-adic solenoids, as well as the universal hyperbolic solenoid of Teichm\"{u}ller theory. Using ideas from shape theory, we classify maps between such solenoids up to homotopy, and we prove a Dehn-Nielsen-type theorem for self-homotopy equivalences of such a solenoid. This generalizes a result of Odden regarding the universal hyperbolic solenoid.