## Topology and Geometric Group Theory Seminar

"Hidden symmetries" of a group G are encoded by its abstract commensurator, the group of isomorphisms between finite-index subgroups of G, modulo equivalence. Just as symmetries of a group G correspond to self-homotopy equivalences of a K(G,1) complex, hidden symmetries of G correspond to homotopy equivalences between finite covers of a K(G,1) complex. When G is of type F, these can all be packaged up as the group of self-homotopy equivalences of the inverse limit of all finite covers of G, a so-called solenoidal space. We will explain these constructions and provide context in classical work of McCord, the use of shape theory, and related dynamical constructions of Sullivan and others. As an application, we will see that any countable union of finite groups can be realized as a group of hidden symmetries of the free group F_2. This work is joint with Edgar A. Bering IV.