Helly groups, coarsely Helly groups, and relative hyperbolicity

TL;DR

This paper explores the properties of Helly and coarsely Helly groups, demonstrating that relative hyperbolicity preserves these properties and establishing geometric structures similar to CAT(0) spaces.

Contribution

It proves that relatively hyperbolic groups with (coarsely) Helly subgroups are themselves (coarsely) Helly, and relates these properties to geometric structures and subgroup behaviors.

Findings

Relatively hyperbolic groups inherit (coarsely) Helly properties from their subgroups.

Classical groups like toral relatively hyperbolic groups act on spaces with convex geodesic bicombing.

Parabolic and quasiconvex subgroups of (coarsely) Helly groups are also (coarsely) Helly.

Abstract

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.