Abstract: The qualitative and quantitative behaviour of simple closed curves on surfaces can reveal a great deal of geometric information about the underlying surface. We look at three theories within this theme, all pertaining to hyperbolic surfaces: Birman and Series's geodesic sparsity theorem, McShane and Rivin's simple length spectrum growth rate asymptotics (as well as later improvements by Mirzakhani), and McShane identities. I hope to give a feel for why these results hold, as well as my input in extending these results to more general types of surfaces structures.
