On denseness of horospheres in higher rank homogeneous spaces
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 $\Gamm...
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| Vydáno v: | Ergodic theory and dynamical systems Ročník 44; číslo 11; s. 3272 - 3289 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
Cambridge, UK
Cambridge University Press
01.11.2024
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| Témata: | |
| ISSN: | 0143-3857, 1469-4417 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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. |
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| Bibliografie: | ObjectType-Article-1 SourceType-Scholarly Journals-1 ObjectType-Feature-2 content type line 14 |
| ISSN: | 0143-3857 1469-4417 |
| DOI: | 10.1017/etds.2024.12 |