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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.16660 |
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| _version_ | 1866910222477426688 |
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| author | Yu, Jing Zhu, Xingyu |
| author_facet | Yu, Jing Zhu, Xingyu |
| contents | We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lovász Local Lemma. For metric measure spaces with volume doubling constant $C_{\mathsf D}$, we prove the sharp bound $\mathrm{asdim}_{AN}(X)\le \mathrm{dim}_{AN}(X)\le \lfloor{\log_2 C_{\mathsf D}}\rfloor$. In particular, if $(M,g)$ is a complete Riemannian $n$-manifold with $\mathrm{Ric}_g\ge 0$, then $\mathrm{asdim}(M)\le n$, thereby settling a question of Papasoglu on manifolds with nonnegative Ricci curvature. We also show that if $(X,\mathsf{d},\mathfrak{m})$ is proper, volume noncollapsed, and has polynomial volume growth rate $ρ^V(X)$, then $\mathrm{asdim}(X)\le \lfloor{ρ^V(X)}\rfloor$. Moreover, the corresponding control function can be chosen to have polynomial growth. This extends Papasoglu's sharp asymptotic-dimension bound from graphs of polynomial growth to a metric-measure setting. As applications, we study equality in the polynomial-growth bound for universal covers of nilmanifolds, and under nonnegative Ricci curvature we relate the equality case in the volume-doubling bound to Gromov largeness, obtaining in particular a consequence for complete manifolds with positive scalar curvature. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_16660 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic-Type Dimension Bounds through Combinatorial Approaches Yu, Jing Zhu, Xingyu Metric Geometry Combinatorics Primary 51F30, Secondary 54F45, 53C23, 05C63 We develop a probabilistic framework for large-scale dimension bounds in metric geometry, based on padded decompositions, randomized ball carving on net graphs, and the Lovász Local Lemma. For metric measure spaces with volume doubling constant $C_{\mathsf D}$, we prove the sharp bound $\mathrm{asdim}_{AN}(X)\le \mathrm{dim}_{AN}(X)\le \lfloor{\log_2 C_{\mathsf D}}\rfloor$. In particular, if $(M,g)$ is a complete Riemannian $n$-manifold with $\mathrm{Ric}_g\ge 0$, then $\mathrm{asdim}(M)\le n$, thereby settling a question of Papasoglu on manifolds with nonnegative Ricci curvature. We also show that if $(X,\mathsf{d},\mathfrak{m})$ is proper, volume noncollapsed, and has polynomial volume growth rate $ρ^V(X)$, then $\mathrm{asdim}(X)\le \lfloor{ρ^V(X)}\rfloor$. Moreover, the corresponding control function can be chosen to have polynomial growth. This extends Papasoglu's sharp asymptotic-dimension bound from graphs of polynomial growth to a metric-measure setting. As applications, we study equality in the polynomial-growth bound for universal covers of nilmanifolds, and under nonnegative Ricci curvature we relate the equality case in the volume-doubling bound to Gromov largeness, obtaining in particular a consequence for complete manifolds with positive scalar curvature. |
| title | Asymptotic-Type Dimension Bounds through Combinatorial Approaches |
| topic | Metric Geometry Combinatorics Primary 51F30, Secondary 54F45, 53C23, 05C63 |
| url | https://arxiv.org/abs/2411.16660 |