The Greatest Common Quotient of Borel-Serre and the Toroidal Compactifications of Locally Symmetric Spaces

Abstract

In this paper, we identify the greatest common quotient (GCQ) of the Borel-Serre compactification and the toroidal compactifications of Hermitian locally symmetric spaces with a new compactification. Using this compactification, we completely settle a conjecture of Harris-Zucker that this GCQ is equal to the Baily-Borel compactification. We also show that the GCQ of the reductive Borel-Serre compactification and the toroidal compactifications is the Baily-Borel compactification. There are two key ingredients in the proof: ergodicity of certain adjoint action on nilmanifolds and incompatibility between the ambient linear structure and the intrinsic Riemannian structure of homothety sections of symmetric cones.