Author ORCID Identifier:

https://orcid.org/0000-0002-6437-3805

Date of Graduation

9-2025

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor/Mentor

Clay, Matt

Committee Member

Matthew Day

Third Committee Member

Yo'av Rieck

Keywords

Cannon-Thurston maps; Free-by-cyclic groups; Geometric group theory; Relatively hyperbolic groups

Abstract

Let F = H1 ∗ · · · ∗ Hk ∗ Fr be a splitting of a finitely generated free group, H = {H1 , . . . , Hk } a set of subgroups of F , and Φ ∈ Aut(F ) an automorphism such that for any i, Φ([Hi ]) = [Hj ] for some j. Consider the free-by-cyclic group Gϕ = ⟨F, t | t−1 xt = Φ(x)⟩. Suppose Gϕ is strongly hyperbolic relative to H̃ = {H̃1 , . . . , H̃k }, where H̃i = ⟨Hi , t | t−1 xt = Φ(x)⟩. In this thesis, we prove the existence and some properties of the Cannon-Thurston map î : ∆(F, H) → ∆(Gϕ , H̃). We build on work of mathematicians studying Cannon-Thurston maps in the contexts of manifolds, hyperbolic group extensions, and hyperbolic free-by-cyclic groups, in particular Cannon and Thurston [CT07], Mj [Mit98], and Kapovich and Lustig [KL15]. Our goal is to consider this work in the context of relatively hyperbolic free-by-cyclic groups, in a similar sense as Pal [Pal10]. Pal proved the existence of the Cannon-Thurston map for certain extensions of relatively hyperbolic groups. We use a construction based on the Grushko trees of splittings, rather than coned-off Cayley graphs, to look at the geometry of relatively hyperbolic free-by-cyclic groups and to prove the existence of the Cannon-Thurston map î. Using this construction, we are able to very quickly observe that î is surjective, and, in a particular case, that the restriction î |∂T : ∂T → ∆Gϕ is surjective as well. We also show that a known theorem about the Cannon-Thurston map for hyperbolic free-by-cyclic groups does not hold in the relative case.

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