Contributions to the essential dimension of finite and algebraic groups

Essential dimension, introduced by Joe Buhler and Zinovy Reichstein and in its most general form by Alexander Merkurjev is a measure of complexity of algebraic objects such as quadratic forms, hermitian forms, central simple algebras and étale algebras. Informally, the essential dimension of an algebraic object is the number of parameters needed to define it.
Often isomorphism classes of objects of some type are in one to one bijection with isomorphism classes of G-torsors. The maximal essential dimension of a G-torsor (called essential dimension of G) gives an invariant of algebraic groups, which will be of primary interest in this thesis. The text is subdivided into four chapters as follows:
Chapter I+II: Multihomogenization of covariants and its application to covariant and essential dimension
The essential dimension of a linear algebraic group G can be expressed via G-equivariant rational maps phi: A(V) --> A(W), so called covariants, between generically free G-modules V and W. In these two chapters we explore a new technique for dealing with covariants, called multihomogenization. This technique was jointly introduced with Hanspeter Kraft and Gerald Schwarz in an already published paper, which forms the second chapter.
Applications of the multihomogenization technique to the essential dimension of algebraic groups are given by results on the essential dimension of central extensions, direct products, subgroups and the precise relation of essential dimension and covariant dimension (which is a variant of the former with polynomial covariants). Moreover the multihomogenization technique allows one to extend a twisting construction introduced by Matthieu Florence from the case of irreducible representations to completely reducible representations. This relates Florence's work on the essential dimension of cyclic p-groups to recent stack theoretic approaches by Patrick Brosnan, Angelo Vistoli and Zinovy Reichstein and by Nikita Karpenko and Alexander Mekurjev.
Chapter III: Faithful and p-faithful representations of minimal dimension
The study of essential dimension of finite and algebraic groups is closely related to the study of its faithful resp. generically free representations. In general the essential dimension of an algebraic group is bounded above by the least dimension of a generically free representation minus the dimension of the algebraic group. In some prominent cases this upper bound or a variant of it is strict.
In this chapter we are guided by the following general questions: What do faithful representations of the least possible dimension look like? How can they be constructed? How are they related to faithful representations of minimal dimension of subgroups? Along the way we compute the minimal number of irreducible representations needed to construct a faithful representation.
Chapter IV: Essential p-dimension of algebraic tori
This chapter is joint work with Mark MacDonald, Aurel Meyer and Zinovy Reichstein. We study a variant of essential dimension which is relative to a prime number p. This variant, called essential p-dimension, disregards effects resulting from other primes than p. In a recent paper Nikita Karpenko and Alexander Merkurjev have computed the essential dimension of p-groups. We extend their result and find the essential p-dimension for a class of algebraic groups, which includes all algebraic tori and twisted finite p-groups.