The title of this thesis refers to the three parts of which it is composed.

The first part concerns the Ekedahl Invariants, new geometric invariants for finite groups introduced in 2009 by Torsten Ekedahl. In Papers A and B, I prove that if G is a subgroup of the three dimensional general linear group over the complex numbers, then the class of its classifying stack is trivial in the Kontsevich value ring of algebraic varieties. This implies that such groups have trivial Ekedahl invariants. If G is a subgroup of the n-dimensional general linear group (over the complex numbers) with abelian reduction in the respective projective linear group, then I show that the Ekedahl invariants satisfy a recurrence relation in a Grothendieck type structure. This relation involves certain cohomologies of the resolution of the singularities of the quotient scheme of the projective space P^n-1 modulo the canonical G action. Finally, I prove that the fifth discrete Heisenberg group has trivial Ekedahl invariants.

The second part of this work focuses on the Veronese modules (Paper C). We extend the results of Bruns and Herzog (about the square free divisor complex) and Paul (about the pile simplicial complex) to the Veronese embeddings and the Veronese modules. We also prove a closed formula for their Hilbert series. Using these results, we study the linearity of the resolution, we characterize when the Veronese modules are Cohen-Macaulay and we give explicit examples of Betti tables of Veronese embeddings.

In the last part of the thesis (Paper D) we prove the existence of linear recurrences of order M with a non-trivial solution vanishing exactly on a subset of the gaps of a numerical semigroup S finitely generated by a₁, a₂, ..., a_N-1, M. This relates to the recent study of linear recurrence varieties by Ralf Fröberg and Boris Shapiro.