Non-collapsed eGH convergence and dimension
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912609488338944 |
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| author | Núñez-Zimbrón, Jesús Santos-Rodríguez, Jaime Zamora, Sergio |
| author_facet | Núñez-Zimbrón, Jesús Santos-Rodríguez, Jaime Zamora, Sergio |
| contents | Let $(X_i,p_i)$ be a non-collapsing sequence of pointed $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and $G_i \leq \text{Iso} (X_i)$ a sequence of closed subgroups of isometries. We show that if the triples $(X_i, G_i, p_i)$ converge in the equivariant Gromov--Hausdorff sense to a triple $(X,G,p)$, then $\text{dim} (G) \geq \limsup _{i \to \infty} \text{dim} (G_i)$, generalizing a result of Harvey to the non-compact setting. The argument also applies in the non-smooth setting of RCD spaces. As an application, we investigate RCD spaces with large isometry groups, extending results of Galaz-García--Kell--Mondino--Sosa and Galaz-García--Guijarro. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-collapsed eGH convergence and dimension Núñez-Zimbrón, Jesús Santos-Rodríguez, Jaime Zamora, Sergio Differential Geometry Metric Geometry 53C23, 53C21 Let $(X_i,p_i)$ be a non-collapsing sequence of pointed $n$-dimensional Riemannian manifolds with a uniform lower Ricci curvature bound, and $G_i \leq \text{Iso} (X_i)$ a sequence of closed subgroups of isometries. We show that if the triples $(X_i, G_i, p_i)$ converge in the equivariant Gromov--Hausdorff sense to a triple $(X,G,p)$, then $\text{dim} (G) \geq \limsup _{i \to \infty} \text{dim} (G_i)$, generalizing a result of Harvey to the non-compact setting. The argument also applies in the non-smooth setting of RCD spaces. As an application, we investigate RCD spaces with large isometry groups, extending results of Galaz-García--Kell--Mondino--Sosa and Galaz-García--Guijarro. |
| title | Non-collapsed eGH convergence and dimension |
| topic | Differential Geometry Metric Geometry 53C23, 53C21 |
| url | https://arxiv.org/abs/2509.22821 |