Geometric structures and Anosov properties for families of representations into $G_2'$

In the exceptional real split Lie group $G_2'$,  Collier and Toulisse introduced cyclic Higgs bundles which have the property of defining representations admitting equivariant holomorphic curves on the pseudosphere. The picture they depict is very similar to that for other families of representations previously constructed in rank 2 Lie groups, which are Anosov and admit associated geometric structures.
In a joint work with Parker Evans we study two families of such representations in $G_2'$. I will  describe these families and present how we construct geometric structures modelled on the $G_2'$ flag manifolds whose holonomies are these representations, using a unifying construction that allows us to reinterpret previously known constructions in other rank 2 Lie groups. I will also present our more recent results about the Anosov properties of these representations.