#### Date of Graduation

5-2020

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy in Mathematics (PhD)

#### Degree Level

Graduate

#### Department

Mathematical Sciences

#### Advisor/Mentor

Matt Clay

#### Committee Member

Matt Day

#### Second Committee Member

Yo'av Rieck

#### Keywords

ascending HNN extensions, endomorphisms, free groups, geometric group theory, Gromov's question, mapping torus

#### Abstract

We prove that ascending HNN extensions of free groups are word-hyperbolic if and only if they have no Baumslag-Solitar subgroups. This extends Brinkmann's theorem that free-by-cyclic groups are word-hyperbolic if and only if they have no Z2 subgroups. To get started on our main theorem, we first prove a structure theorem for injective but nonsurjective endomorphisms of free groups. With the decomposition of the free group given by this structure theorem, we (more or less) construct representatives for nonsurjective endomorphisms that are expanding immersions relative to a homotopy equivalence. This structure theorem initializes the development of (relative) train track theory for nonsurjective endomorphisms.

#### Citation

Mutanguha, J.
(2020). Hyperbolic Endomorphisms of Free Groups. * Graduate Theses and Dissertations*
Retrieved from https://scholarworks.uark.edu/etd/3641