Cartan projections of some non-reductive subgroups and proper actions on homogeneous spaces

TL;DR

This paper extends Kobayashi's method using Cartan projections to analyze non-reductive subgroups, providing new criteria and examples for the existence of compact Clifford-Klein forms on certain homogeneous spaces.

Contribution

It generalizes Kobayashi's approach to non-reductive subgroups, offering new tools and examples for understanding proper actions on homogeneous spaces.

Findings

Extended Kobayashi's method to non-reductive subgroups

Identified homogeneous spaces that do not admit compact Clifford-Klein forms

Compared Cartan projections of reductive and non-reductive subgroups

Abstract

Kobayashi [Duke Math. J. (1992)] gave a necessary condition for the existence of compact Clifford-Klein forms in terms of Cartan projections and non-compact dimensions of reductive subgroups. We extend his method to non-reductive subgroups, and give some examples of homogeneous spaces of reductive type that do not admit compact Clifford-Klein forms by comparing Cartan projections and non-compact dimensions of reductive subgroups with those of non-reductive subgroups.