Discrete Group Actions and Spectral Geometry of Crossed Products

The (twisted) crossed product construction is fundamental in the theory of $C^*$-algebras and in noncommutative topology since it represents the operation of forming a quotient when this is a singular, badly-behaved space. For instance, the study of noncommutative coverings, in the special case of finite abelian structure groups, shows that twisted crossed products are the noncommutative analogue of topological regular coverings. Since spectral triples are a central notion in noncommutative geometry, this makes the task of constructing spectral triples on crossed products a natural subject of interest. In this thesis, we construct and study spectral triples on reduced twisted crossed products $A times^{sigma}_{alpha,r}G$, where $A$ is a unital $C^*$-algebra, $G$ a discrete group and $(alpha,sigma)$ a twisted action in the sense of Busby and Smith cite{busby1970representations}. For this construction we follow, as in cite{hawkins2013spectral}, the guiding principle of the Kasparov external product, combining the given Dirac operator on $A$ with a matrix valued length-type function on the group. In particular, we provide sufficient conditions so that this triple on $A times_{alpha,r}^{sigma}G$ satisfies some of the {em axioms} of noncommutative manifolds cite{connes1996gravity}: summability, regularity, compatibility with real structures, first and second order conditions and orientation cycles. Our guide example is the spectral triple on the noncommutative $2$-torus cite{rieffel1981c,gracia2013elements}, regarded as the crossed product $C(S^{1}) times mathbb{Z}$. We show that our constructions generalize the usual ones on its triple.