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Résumé |
In this talk we will consider two families of representations from the fundamental group of a closed surface of genus at least 2 into the exceptional Lie group G_2, and more precisely into its real split form G'_2. Representations in these families correspond to Higgs bundles of a very special form introduced by Collier and Toulisse. They come with associated equivariant objects: they admit an alternating almost-complex map into the pseudosphere S^2,4, which can be reinterpreted as a parallel distribution of planes along a minimal surface in the symmetric space.
From the Higgs bundle description of these families, however, it is far from clear whether these representations have good geometric properties. In joint work with Parker Evans, we use the equivariant objects to construct explicitly a geometric structure associated to some of these representations.
After an introduction to the geometry of G'_2 and to these two families of representations, I will present our results explaining how to construct for every representation \rho; in the first family a geometric structure modelled on a flag manifold of G2, the Einstein universe Ein^2,3, whose holonomy is \rho. This is a structure on a fiber bundle over the considered surface with fiber diffeomorphic to Ein^2,1.
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