# Domains of discontinuity of Anosov representations in flag manifolds and oriented flag manifolds

An infinite discrete subgroup of a Lie group acts on its homogeneous spaces. If the action is proper on an open subset, we call this subset a domain of discontinuity. In this thesis we investigate criteria when this happens, for some groups and spaces.

In the first part, we consider the action of an Anosov subgroup Γ ⊂ G of a semi–simple Lie group on the associated flag manifolds. It is known that domains of discontinuity can be constructed from combinatorial objects called balanced ideals [KLP18]. For ∆–Anosov groups, we prove that every maximal and every cocompact domain of discontinuity arises from this construction, up to a few exceptions in low rank. In particular, this shows that some flag manifolds admit no cocompact domain of discontinuity. Applied to Hitchin rep- resentations, we determine exactly those flag manifolds which admit cocompact domains of discontinuity and give the number of different domains in the case of Grassmannians.

In the second part, we extend the theory of balanced ideals to the action of Γ ⊂ G on oriented flag manifolds. These are quotients G/P, where P is a subgroup lying between a parabolic subgroup and its identity component. Under the condition that the limit curve of Γ lifts to some oriented flag manifold, we identify cocompact domains of discontinuity in oriented flag manifolds which we do not see in the unoriented setting. They even exist in some cases where in the unoriented flag manifold there are no cocompact domains at all. These include in particular domains in some oriented Grassmannians for Hitchin representations, which we also show to be nonempty.

As another application of the oriented setup, we give a new lower bound on the number of connected components of ∆–Anosov representations of a closed surface group into SL(n, R). We further use certain balanced ideals to construct a compactification of locally symmetric spaces arising from Anosov representations into Sp(2n, R). Finally, we discuss an approach to generalize the construction of domains of discontinuity to other homogeneous spaces.

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