Teichmüller space can be viewed as the space of marked conformal structures or, equivalently, marked hyperbolic structures on a surface. Topologically, it is homeomorphic to an open cell that admits natural metrics and compactifications from both perspectives: namely, the Teichmüller metric and the Gardiner-Masur compactification in the conformal viewpoint, and the Thurston metric or arc metric and the Thurston compactification in the hyperbolic viewpoint. A central theme established in recent decades is that for closed surfaces, these compactifications coincide naturally with the horofunction compactifications of the corresponding metrics. This thesis investigates these boundary structures and extends these foundational equivalences to broader classes of surfaces.

In the first part of the thesis, we focus on the conformal setting, and the Gardiner-Masur compactification. A point in the Gardiner-Masur boundary is called a Busemann point if it arises as the limit of an almost-geodesic ray. Addressing a question regarding the density of such points, Azemar proved that the set of Busemann points is not dense in the Gardiner-Masur boundary of the Teichmüller space of surfaces without boundary (of genus $g$ with $n$ punctures). In this thesis, we extend Azemar’s non-density result to the Teichmüller space of bordered surfaces. Our proof employs a novel, unified approach based on extremal length identities; this technique applies equally to surfaces without boundary, thereby providing an alternative proof of Azemar’s result.

In the second part of the thesis, we shift to the hyperbolic perspective and develop an analogous arc metric for the Teichmüller space of crowned hyperbolic surfaces (uniformized bordered surfaces with marked points on the boundary). We extend the results of Walsh and Alessandrini-Liu-Papadopoulos-Su by establishing that the horofunction compactification of the Teichmüller space of crowned and bordered hyperbolic surfaces with respect to this arc metric is homeomorphic to its Thurston compactification. At the core of our proof lies an asymptotic estimation of arc lengths along Thurston stretch rays. The previous known methods do not apply for crowned hyperbolic surfaces because their boundary is non-compact. Hence, to obtain these estimates on crowned hyperbolic surfaces, we adopt an alternative approach based on dual $\mathbb{R}$-trees associated with measured laminations. This yields an independent proof of the bordered case.