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