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There are two closely related classes of groups arising from Fuchsian groups and their actions on the hyperbolic plane \( \H^2 \): discrete subgroups of semisimple Lie groups acting on symmetric spaces, and mapping class groups and their subgroups acting on Teichmüller space. Convex cocompact Fuchsian groups have been generalized to Anosov subgroups of semisimple Lie groups of higher rank, which have played an important role in higher Teichmüller theory. They have also been generalized to convex cocompact subgroups of mapping class groups. Due to the fact that Teichmüller space shares some similarities with symmetric spaces of rank one, the analogy between convex cocompact Fuchsian groups and convex cocompact subgroups of mapping class groups is rather complete. But less is known about actions of Anosov subgroups on symmetric spaces of higher rank.
In this chapter, we discuss three conjectures on compactifications and coarse fundamental domains for locally symmetric spaces associated with Anosov subgroups of noncompact semisimple Lie groups, and describe results, motivations, and evidence for these conjectures. One conjecture deals with the existence of maximal Satake compactifications of Anosov locally symmetric spaces, and other two are concerned with the characterization of Anosov subgroups. By comparing these locally symmetric spaces of infinite volume with symmetric spaces and locally symmetric spaces of finite volume, and by examining applications of the maximal Satake compactification of symmetric spaces and the Borel-Serre compactification of locally symmetric spaces of finite volume, we conclude that the conjectural maximal Satake compactifications of Anosov locally symmetric spaces arising from the maximal Satake compactification of symmetric spaces are the natural compactifications. In general, there is more than one maximal Satake compactification for Anosov locally symmetric spaces. To explain this non-uniqueness, we develop a reduction theory for Anosov subgroups by introducing the crucial notion of anti-Siegel sets. We also explain how geometric boundaries of the maximal Satake compactifications are related to problems in the spectral theory of Laplace operators of the Riemannian manifolds under consideration, and conclude with some questions on the Martin compactification of Anosov locally symmetric spaces, which is related to positive harmonic functions.