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On denseness of horospheres in higher rank homogeneous spaces
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 44 / Issue 11 / November 2024
- Published online by Cambridge University Press:
- 19 February 2024, pp. 3272-3289
- Print publication:
- November 2024
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- Article
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Let
$ G $ be a connected semisimple real algebraic group and 
$\Gamma <G$ be a Zariski dense discrete subgroup. Let N denote a maximal horospherical subgroup of G, and 
$P=MAN$ the minimal parabolic subgroup which is the normalizer of N. Let 
$\mathcal E$ denote the unique P-minimal subset of 
$\Gamma \backslash G$ and let 
$\mathcal E_0$ be a 
$P^\circ $-minimal subset. We consider a notion of a horospherical limit point in the Furstenberg boundary 
$ G/P $ and show that the following are equivalent for any 
$[g]\in \mathcal E_0$: (1)
$gP\in G/P$ is a horospherical limit point;(2)
$[g]NM$ is dense in 
$\mathcal E$;(3)
$[g]N$ is dense in 
$\mathcal E_0$.
The equivalence of items (1) and (2) is due to Dal’bo in the rank one case. We also show that unlike convex cocompact groups of rank one Lie groups, the
$NM$-minimality of 
$\mathcal E$ does not hold in a general Anosov homogeneous space.
