Helly groups, coarsely Helly groups, and relative hyperbolicity

A simplicial graph is said to be ( coarsely ) Helly if any collection of pairwise intersecting balls has non-empty (coarse) intersection. ( Coarsely ) Helly groups are groups acting geometrically on (coarsely) Helly graphs. Our main result is that finitely generated groups that are hyperbolic relative to (coarsely) Helly subgroups are themselves (coarsely) Helly. One important consequence is that various classical groups, including toral relatively hyperbolic groups, are equipped with a CAT(0)-like structure—they act geometrically on spaces with convex geodesic bicombing. As a means of proving the main theorems we establish a result of independent interest concerning relatively hyperbolic groups: a ‘relatively hyperbolic’ description of geodesics in a graph on which a relatively hyperbolic group acts geometrically. In the other direction, we show that for relatively hyperbolic (coarsely) Helly groups their parabolic subgroups are (coarsely) Helly as well. More generally, we show that ‘quasiconvex’ subgroups of (coarsely) Helly groups are themselves (coarsely) Helly.