Topological rigidity of small RCD(K,N) spaces with maximal rank
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2024
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866915693053607936 |
|---|---|
| author | Zamora, Sergio Zhu, Xingyu |
| author_facet | Zamora, Sergio Zhu, Xingyu |
| contents | For a polycyclic group $Λ$, $\text{rank} (Λ)$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $Λ$. For a finitely generated group $G$, $\text{rank} (G)$ is defined as the infimum of $ \text{rank} (Λ)$ among finite index polycyclic subgroups $Λ\leq G$.
For a compact $ \text{RCD} (K,N)$ space $(X,\mathsf{d}, \mathfrak{m})$ with $ \text{diam} (X) \leq \varepsilon (K,N)$, the rank of $π_1(X)$ is at most $N$. We show that in case of equality, $X$ is homeomorphic to an infranilmanifold, generalizing a result by Kapovitch--Wilking to the non-smooth setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10189 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological rigidity of small RCD(K,N) spaces with maximal rank Zamora, Sergio Zhu, Xingyu Differential Geometry Metric Geometry 53C23, 53C21 For a polycyclic group $Λ$, $\text{rank} (Λ)$ is defined as the number of $\mathbb{Z}$ factors in a polycyclic decomposition of $Λ$. For a finitely generated group $G$, $\text{rank} (G)$ is defined as the infimum of $ \text{rank} (Λ)$ among finite index polycyclic subgroups $Λ\leq G$. For a compact $ \text{RCD} (K,N)$ space $(X,\mathsf{d}, \mathfrak{m})$ with $ \text{diam} (X) \leq \varepsilon (K,N)$, the rank of $π_1(X)$ is at most $N$. We show that in case of equality, $X$ is homeomorphic to an infranilmanifold, generalizing a result by Kapovitch--Wilking to the non-smooth setting. |
| title | Topological rigidity of small RCD(K,N) spaces with maximal rank |
| topic | Differential Geometry Metric Geometry 53C23, 53C21 |
| url | https://arxiv.org/abs/2406.10189 |