Exotic proper actions on homogeneous spaces via convex cocompact representations

TL;DR

This paper constructs specific homogeneous spaces that admit proper actions by certain discrete subgroups but not by non-compact semisimple subgroups, expanding understanding of group actions on homogeneous spaces.

Contribution

It introduces new examples of homogeneous spaces with unique proper action properties, combining convex cocompact representations and nilpotent orbit theory.

Findings

Existence of homogeneous spaces with proper discrete subgroup actions but no non-compact semisimple subgroup actions.

Construction based on convex cocompact subgroups of O(n,1) and Coxeter groups.

Non-existence proved using nilpotent orbit theory and combinatorics.

Abstract

We construct a series of homogeneous spaces G/H of reductive type which admit proper actions of discrete subgroups of G isomorphic to cocompact lattices of O(n,1) (n=2,3,4) but do not admit proper actions of non-compact semisimple subgroups of G. The existence of such homogeneous spaces was previously not known even for n=2. Our construction of proper actions of discrete subgroups is based on Gu\'eritaud-Kassel's work on convex cocompact subgroups of O(n,1) and Danciger-Gu\'eritaud-Kassel's work on right-angled Coxeter groups. On the other hand, the non-existence of proper actions of non-compact semisimple subgroups is proved by the theory of nilpotent orbits and elementary combinatorics.