Transverse groups preserving proper domains in flag manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866909698372927488 |
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| author | Galiay, Blandine |
| author_facet | Galiay, Blandine |
| contents | 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--Guéritaud--Kassel. This result highlights the spatial nature of the dynamics of $H$. We construct Zariski-dense examples of such transverse subgroups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_15891 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Transverse groups preserving proper domains in flag manifolds Galiay, Blandine Representation Theory Differential Geometry Group Theory 20F65, 22E40, 57S25 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--Guéritaud--Kassel. This result highlights the spatial nature of the dynamics of $H$. We construct Zariski-dense examples of such transverse subgroups. |
| title | Transverse groups preserving proper domains in flag manifolds |
| topic | Representation Theory Differential Geometry Group Theory 20F65, 22E40, 57S25 |
| url | https://arxiv.org/abs/2507.15891 |