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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| Materias: | |
| Acceso en línea: | https://arxiv.org/abs/2504.01788 |
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| _version_ | 1866912305881546752 |
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| author | Bucher, Michelle Savini, Alessio |
| author_facet | Bucher, Michelle Savini, Alessio |
| contents | Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in G/P$ is opposite to the Weyl chambers determined by $F$, we define a projection $Φ(F, x) \in F$ which is continuous and $G$-equivariant. Furthermore, if $q \geq 3$, we exhibit a $G$-equivariant continuous map defined on an open subset of full measure of the space of $q$-tuples of $(G/P)^q$ with image in $G/K$. When $G$ is the orientation preserving isometries of real hyperbolic space and $q = 3$, we recover the geometric barycenter of the corresponding ideal triangle. All our proofs are constructive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_01788 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Projections from Furstenberg boundaries onto maximal flats and barycenter maps Bucher, Michelle Savini, Alessio Group Theory Differential Geometry 22F30, 57S20 Let $G$ be a semisimple connected Lie group of non-compact type with finite center. Let $K<G$ be a maximal compact subgroup and $P<G$ be a minimal parabolic subgroup. For any pair $(F,x)$, where $F$ is a maximal flat in $G/K$ and $x \in G/P$ is opposite to the Weyl chambers determined by $F$, we define a projection $Φ(F, x) \in F$ which is continuous and $G$-equivariant. Furthermore, if $q \geq 3$, we exhibit a $G$-equivariant continuous map defined on an open subset of full measure of the space of $q$-tuples of $(G/P)^q$ with image in $G/K$. When $G$ is the orientation preserving isometries of real hyperbolic space and $q = 3$, we recover the geometric barycenter of the corresponding ideal triangle. All our proofs are constructive. |
| title | Projections from Furstenberg boundaries onto maximal flats and barycenter maps |
| topic | Group Theory Differential Geometry 22F30, 57S20 |
| url | https://arxiv.org/abs/2504.01788 |