On the Coarse Lusternik-Schnirelmann Category of Groups
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arXiv
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915612600565760 |
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| author | De Saha, Aditya |
| author_facet | De Saha, Aditya |
| contents | We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by $\text{c-cat}$, defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological properties of groups and spaces. We establish that $\text{c-cat}$ is a coarse homotopy invariant and prove a lower-bound $\text{p-cat}(Γ)\leq \text{c-cat}(Γ)$ for geometrically finite groups $Γ$, where $\text{p-cat}$ denotes the proper LS-category introduced in 1992 by Ayala and co-authors. We also prove an upper bound $\text{c-cat}(Γ) \leq \text{asdim}(Γ)$ for bicombable 1-ended groups which are semistable at $\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_10367 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Coarse Lusternik-Schnirelmann Category of Groups De Saha, Aditya Geometric Topology Algebraic Topology Group Theory Metric Geometry 20F65 (Primary) 51F30, 55M30 (Secondary) We introduce a coarse analog of the classical Lusternik-Schnirelmann category which we denote by $\text{c-cat}$, defined for metric spaces in the coarse homotopy category. This provides a new tool for studying large-scale topological properties of groups and spaces. We establish that $\text{c-cat}$ is a coarse homotopy invariant and prove a lower-bound $\text{p-cat}(Γ)\leq \text{c-cat}(Γ)$ for geometrically finite groups $Γ$, where $\text{p-cat}$ denotes the proper LS-category introduced in 1992 by Ayala and co-authors. We also prove an upper bound $\text{c-cat}(Γ) \leq \text{asdim}(Γ)$ for bicombable 1-ended groups which are semistable at $\infty$. |
| title | On the Coarse Lusternik-Schnirelmann Category of Groups |
| topic | Geometric Topology Algebraic Topology Group Theory Metric Geometry 20F65 (Primary) 51F30, 55M30 (Secondary) |
| url | https://arxiv.org/abs/2510.10367 |