The Thurston Boundary of Hyperbolic Knot Complements

This paper explores the Thurston boundary, a crucial compactification of the Teichmüller space, specifically in the context of hyperbolic knot complements. William P. Thurston's groundbreaking work in the late 1970s and early 1980s revolutionized the understanding of 3-manifolds by establishing that most knot complements admit hyperbolic structures. The Thurston boundary provides a natural and geometrically significant compactification, where points at infinity correspond to projective classes of measured geodesic laminations on surfaces. We delve into the theoretical foundations, starting with hyperbolic geometry in three dimensions and the classification of 3-manifolds via Thurston's Geometrization Conjecture. The knot complement, typically a non-compact 3-manifold with a torus boundary, becomes the focus. We investigate how sequences of hyperbolic metrics on these complements can degenerate, leading to points on the Thurston boundary. The concept of measured laminations, which are central to defining this boundary, is meticulously examined. Furthermore, the paper discusses the role of Kleinian groups in uniformizing hyperbolic 3-manifolds and how their limit sets relate to the boundary. Special attention is paid to the implications for understanding the asymptotic behavior of hyperbolic structures on knot complements and the connection to Cannon-Thurston maps. The study aims to elucidate the intricate relationship between the topology of knot complements and their geometric compactifications, offering insights into the broader field of low-dimensional topology and geometric group theory.