On the Hirzebruch-Kobayashi-Ono proportionality principle and the non-existence of compact solvable Clifford-Klein forms of certain homogeneous spaces

Abstract

This article continues a line of research aimed at solving an important problem of T. Kobayashi of the existence of compact Clifford-Klein forms of reductive homogeneous spaces. We contribute to this topic by showing that almost all symmetric spaces and 3-symmetric spaces do not admit solvable compact CliffordfKlein forms (with several possible exceptions). Our basic tool is a combination of the Hirzebruch-Kobayashi-Ono proportionality principle with the theory of syndetic hulls. Using this, we prove a general theorem which yields a sufficient condition for the non-existence of compact solvable CliffordKlein forms.