Submodular functions in additive combinatorics problems for group actions and representations

Abstract

This article deals with generalisations of some classical problems and results
in additive combinatorics of groups to the context of group actions or group
representations. We show that the classical methods are sufficiently deep to
extend to this wider context where, instead of two free transitive commuting
actions (left and right multiplications on the group), there is only one single
action. Following ideas of Hamidoune and Tao, our main tool is the notion of
G-invariant submodular function defined on power sets. We are able to extend
to this group action context results of Hamidoune and Tao as well as results of
Murphy and Ruzsa.