In this thesis we give a construction of cyclic cocycles on convolution
algebras twisted by gerbes over discrete translation groupoids. In his seminal
book, Connes constructs a map from the equivariant cohomology of a manifold
carrying the action of a discrete group into the periodic cyclic cohomology of
the associated convolution algebra. Furthermore, for proper \'etale groupoids,
J.-L. Tu and P. Xu provide a map between the periodic cyclic cohomology of a
gerbe twisted convolution algebra and twisted cohomology groups. Our focus will
be the convolution algebra with a product defined by a gerbe over a discrete
translation groupoid. When the action is not proper, we cannot construct an
invariant connection on the gerbe; therefore to study this algebra, we instead
develop simplicial notions related to ideas of J. Dupont to construct a
simplicial form representing the Dixmier-Douady class of the gerbe. Then by
using a JLO formula we define a morphism from a simplicial complex twisted by
this simplicial Dixmier-Douady form to the mixed bicomplex of certain matrix
algebras. Finally, we define a morphism from this complex to the mixed
bicomplex computing the periodic cyclic cohomology of the twisted convolution
algebras.Comment: PhD thesis, 87 page