** Title: Hyperbolic Endomorphisms of Free Groups. **

*Time*: Tuesday 14 April 2020 at 2PM CDT.

*Location*: Remote. Contact me (email, twitter, best way you know how) for the link.

*Abstract*: In my thesis, I prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends Brinkmannâ€™s theorem that free-by-cyclic groups are word-hyperbolic if and only if they have no \(\mathbb Z^2\) subgroups. The first part of this talk will be an introduction into the subject in which I will define all the terms in the statement of the main theorem and explain the motivation behind the theorem. In the second part, I will discuss a structure theorem for endomorphisms of free groups that got me started on proving the main theorem. There will be examples to illustrate how this structure theorem allows me to (more or less) construct representatives for nonsurjective endomorphisms that are expanding immersions relative to a homotopy equivalence.

*Last updated: Fri 10 April 2020*