Transverse groups preserving proper domains in flag manifolds

TL;DR

This paper investigates conditions under which subgroups of semisimple Lie groups preserve proper domains in flag manifolds, introducing causal convexity in the Hermitian case and exploring the dynamics of transverse subgroups.

Contribution

It establishes a necessary condition for subgroup preservation of domains, introduces causal convexity in the Hermitian setting, and constructs examples of transverse subgroups with specific dynamical properties.

Findings

Subgroups transverse to a parabolic preserve certain geometric properties.

Causal convexity relates to subgroup dynamics in Hermitian symmetric spaces.

Zariski-dense examples of such subgroups are constructed.

Abstract

Given a semisimple Lie group $G$ and a self-opposite flag manifold $\mathcal{F}$ of $G$, we establish a necessary condition for an infinite subgroup $H$ of $G$ to preserve a proper domain in $\mathcal{F}$. In the case where $G$ is a Hermitian Lie group of tube type, we introduce and study a notion of causal convexity in the Shilov boundary $\mathbf{Sb}(G)$ of the symmetric space of $G$, inspired by the one already existing in conformal Lorentzian geometry. We show that subgroups $H$ of $G$ that are transverse with respect to a parabolic subgroup of $G$ defining $\mathbf{Sb}(G)$ and that preserve a proper domain in $\mathbf{Sb}(G)$ satisfy a geometric property with respect to this causal convexity, close to the strong projective convex cocompactness defined by Danciger--Gu\'eritaud--Kassel. This result highlights the spatial nature of the dynamics of $H$. We construct Zariski-dense examples of such transverse subgroups.