This thesis is mainly concerned with the study of the crossed product $C^*$-algebras associated to the horocycle flow on compact quotients of $\SL(2,\rr)$. Looking at the Cuntz semigroup, we retrieve some information about the structure of hereditary $C^*$-subalgebras and Hilbert modules for a class of $C^*$-algebras which contain the $C^*$-algebras we want to study. After translating these results in our context, we study the functoriality of the construction both for the case of discrete subgroups of $\SL(2,\rr)$ and for the case of hyperbolic Riemann surfaces. Also properties of another crossed product $C^*$-algebra that is Morita equivalent to the $C^*$-algebra of the horocycle flow are explored and from considerations about the associated dynamical system we can prove that the multiplier algebra of the crossed product $C^*$-algebra associated to the horocyce flow contains a Kirchberg algebra in the UCT class as a unital $C^*$-subalgebra in some cases.\\
A side chapter is devoted to a project that the author started during his PhD, concerning the construction of spectral triples on the Jiang-Su algebra. The construction we give is performed by means of a particular $AF$-embedding.
