Thesis advisor: Ian Biringer / We consider the space $\rootedH2$ of all complete hyperbolic surfaces without boundary with a basepoint equipped with the pointed Gromov-Hausdorff topology. Continuous paths within $\rootedH2$ arising from certain deformations on a hyperbolic surface and concrete geometric constructions are studied. These include changing some Fenchel-Nielsen parameters of a subsurface, pinching a simple closed geodesic to a cusp, and inserting an infinite strip along a proper bi-infinite geodesic. We then use these paths to show that $\rootedH2$ is path-connected and that it is locally weakly connected at points whose underlying surfaces are either the hyperbolic plane or hyperbolic surfaces of the first kind. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Mathematics.

京都大学 / 新制・課程博士 / 博士(理学) / 甲第22972号 / 理博第4649号 / 新制||理||1668(附属図書館) / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 藤原 耕二, 教授 山口 孝男, 教授 入谷 寛 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM