On the Coarse Lusternik-Schnirelmann Category of Groups

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 $\text{p-cat}(\Gamma)\leq \text{c-cat}(\Gamma)$ for geometrically finite groups $\Gamma$, where $\text{p-cat}$ denotes the proper LS-category introduced in 1992 by Ayala and co-authors. We also prove an upper bound $\text{c-cat}(\Gamma) \leq \text{asdim}(\Gamma)$ for bicombable 1-ended groups which are semistable at $\infty$.