Properly Discontinuous Actions on Homogeneous Spaces

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The main motivation for this project is the so-called Auslander's conjecture, which is a very fundamental question about groups of affine transformations that act properly and discontinuously on the affine space. These groups roughly correspond to regular affine tilings, i.e., tilings where all the tiles have the same shape up to an affine transformation. These groups also admit an interpretation in differential geometry as flat affine manifolds. The special case where these groups preserve a Euclidean metric, which in the language of tilings means that all the tiles have the same shape in an ordinary sense, has been very well-known for over a century: this is basically the subject of classical crystallography. In the general case, there is a much larger variety of these groups; and the question of their classification also leads to some fascinating questions in the theory of Lie groups and their representations.The first part of the project consists in working on a conjecture that would give, for every semisimple real Lie group G and irreducible representation V thereof, a necessary and sufficient criterion for the existence of a subgroup Gamma in the group of affine automorphisms of V with linear part in G, such that Gamma has linear part Zariski-dense in G, is free nonabelian and acts properly discontinuously on V. The second part of the project consists in translating this abstract algebraic criterion into a simple condition on the highest restricted weight of the representation, and thus completely classifying Zariski-closures of such subgroups. The last part of the proposal consists in gaining a better understanding of these groups: for instance, constructing whenever possible a fundamental domain corresponding to their proper action on the affine space; going beyond proving existence or non-existence of such groups, by looking for results that classify all such groups for a given Zariski closure; and trying to link them with free groups acting properly on other homogeneous spaces.