Taylor, Samuel J.2024-09-122024-09-122024-08http://hdl.handle.net/20.500.12613/10682An overarching question in the study of 3-manifolds is that of how a given 3-manifold fibers over the circle. Thurston and Fried investigated this question for a closed hyperbolic 3-manifold M. They found an open fibered cone in H^1(MR) for which each integral point corresponds to a fibration of M over S^1. Landry, Minsky, and Taylor extended the investigation of the fibered cone to its boundary. They found infinite-type surfaces with end-periodic maps naturally occurring in the boundary. Dowdall, Kapovich, and Leininger investigated fibrations over the circle for the mapping torus of an irreducible train track map on a finite graph with no valence-1 vertices. They found an open fibered cone in the first cohomology of the mapping torus for which each integral point corresponds to a fibration of the mapping torus over S^1. The research conducted in this thesis lives in this setting. We will address the natural question of the significance of the boundary of the fibered cone for this mapping torus. We construct finite graphs which naturally arise in the boundary of the fibered cone. We then use these finite graphs to construct infinite graphs in the boundary with natural maps with properties analogous to those of end-periodic surface homeomorphisms.123 pagesengIN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available.http://rightsstatements.org/vocab/InC/1.0/MathematicsText157792024-08-30MeadowMacLeod_temple_0225E_15779.pdf