Helly groups, coarsely Helly groups, and relative hyperbolicity (Q6567113)

Let \(\Gamma\) be a simplicial graph with vertex set equipped with the combinatorial metric. The graph \(\Gamma\) is Helly (coarsely Helly) if given any collection of balls \(\{B_{\rho_{i}}(x_{i}) \mid i \in \mathcal{I} \}\) in \(\Gamma\) such that \(B_{\rho_{i}}(x_{i}) \cap B_{\rho_{j}}(x_{j}) \not = \emptyset\) for every \(i,j \in \mathcal{I}\) then \(\bigcap_{i \in \mathcal{I}}B_{\rho_{i}}(x_{i}) \not = \emptyset\) (\(\bigcap_{i \in \mathcal{I}}B_{\rho_{i}+\xi}(x_{i}) \not = \emptyset\) for some universal constant \(\xi \geq 0\)). A group is Helly (coarsely Helly) if it acts geometrically on a Helly (coarsely Helly) graph.\N\NThe main results in this paper are Theorem 1.1 and Theorem 1.2: Let \(G\) be a finitely generated group that is hyperbolic relative to a collection of Helly subgroups (coarsely Helly subgroups). Then \(G\) is Helly (coarsely Helly).\N\NAn important consequence is that various classical groups, including toral relatively hyperbolic groups, are equipped with a \(\mathsf{CAT}(0)\)-like structure.\N\NIn the other direction, the authors show that for relatively hyperbolic (coarsely) Helly groups their parabolic subgroups are (coarsely) Helly as well (Theorem 1.3). More generally, they show that quasiconvex subgroups of (coarsely) Helly groups are themselves (coarsely) Helly (Theorem 1.4).\N\NTheorem 1.4 can be used to construct new examples of groups that are not (coarsely) Helly. For instance, \textit{N. Hoda} [Bull. Lond. Math. Soc. 55, No. 6, 2991--3011 (2023; Zbl 1529.20074)] characterised virtually nilpotent Helly groups, showing that they are all virtually abelian and act geometrically on \((\mathbb{R}^{n}, ||\cdot ||_{\infty}))\) for some \(n\). Hence, if \(G\) is a group and \(H < G\) is a strongly quasiconvex virtually nilpotent subgroup not satisfying these properties, then \(G\) is not Helly.