Perfect Subgroups of Hnn Extensions

This note includes supporting material for Guilbault’s one-hour talk summarized elsewhere in these proceedings. We supply the group theory necessary to argue that Guilbault’s tame ends cannot be pseudocollared. In particular, we show that certain groups (the interated Baumslag-Solitar groups) cannot have any non-trivial perfect subgroups. The absence of non-trivial perfect subgroups, in turn, eliminates the possibility of non-trivial homotopy equivalences. In contrast, we include an example of a pseudo-collared end based on groups (the interated Adam’s groups) that are somewhat similar to the Baumslag-Solitar groups. We close with a discussion of a homotopy theoretic approach to this construction.