Geometry of Satake and Toroidal Compactifications

In [JM02, section 14], Ji and MacPherson give new constructions of the Borel--Serre and reductive Borel--Serre compactifications [BS73, Zuc82] of a locally symmetric space. They use equivalence classes of eventually distance minimizing (EDM) rays to describe the boundaries of these compactications. The primary goal of this thesis is to construct the Satake compactifications of a locally symmetric space [Sat60a] using finer equivalence relations on EDM rays. To do this, we first construct the Satake compactifications of the global symmetric space [Sat60b] with equivalence classes of geodesics in the symmetric space. We then define equivalence relations on EDM rays using geometric properties of their lifts in the symmetric space. We show these equivalence classes are in one-to-one correspondence with the points of the Satake boundary. As a secondary goal, we outline the construction of the toroidal compactifications of Hilbert modular varieties [Hir71, Ehl75] using a larger class of toric curves and equivalence relations that depend on the compactications\u27 defining combinatorial data