Homogeneous space with non virtually abelian discontinuous groups but without any proper SL(2,R)-action

TL;DR

This paper constructs a counterexample demonstrating that certain homogeneous spaces of reductive type admit non virtually abelian discontinuous groups without supporting any proper SL(2,R)-action, challenging previous assumptions.

Contribution

It provides the first known counterexample in the case G=SL(5,R), showing the limits of extending previous theorems about proper SL(2,R)-actions.

Findings

Counterexample in SL(5,R) case

Discontinuous groups not approximated by connected groups

Limits of previous theorems on homogeneous spaces

Abstract

In the study of discontinuous groups for non-Riemannian homogeneous spaces, the idea of "continuous analogue" gives a powerful method (T. Kobayashi [Math. Ann. 1989]). For example, a semisimple symmetric space G/H admits a discontinuous group which is not virtually abelian if and only if G/H admits a proper SL(2,R)-action (T. Okuda [J. Differential Geom. 2013]). However, the action of discrete subgroups is not always approximated by that of connected groups. In this paper, we show that the theorem cannot be extended to general homogeneous spaces G/H of reductive type. We give a counterexample in the case G = SL(5,R).