Stable properties of plethysm: on two conjectures of Foulkes.
It is not known whether or not the stable rational cohomology groups H*(Aut(F∞);Q) always vanish (see Hatcher in [5] and Hatcher and Vogtmann in [7] where they pose the question and show that it does vanish in the first 6 dimensions). We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields...
We start with a small paradigm shift about group representations, namely the observation that restriction to a subgroup can be understood as an extension-of-scalars. We deduce that, given a group , the derived and the stable categories of representations of a subgroup can be constructed out of the corresponding category for by a purely triangulated-categorical construction, analogous to étale extension in algebraic geometry. In the case of finite groups, we then use descent methods to investigate...
Let be a non-trivial algebraically closed group and be a subset of generating in infinitely many steps. We give a construction of a binary tree associated with . Using this we show that if is -existentially closed then it is strongly bounded.