Helly groups, coarsely Helly groups, and relative hyperbolicity

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.

Figures (12)

• Figure 1: The proof of Proposition \ref{['prop:BCPtriangles']}.
• Figure 2: The quotient graph $\widetilde{\Gamma}(N)/F$. Orbits of medial vertices are shown in green, connecting edges in blue, free vertices and edges in gray. Orbits of internal vertices and edges are represented by the red regions.
• Figure 3: An example construction of a derived path $\widehat{P} \subseteq \operatorname{Cay}(G,X \cup \mathcal{H})$ given a path $P \subseteq \Gamma(N)$. In this case, $|P'| \geq 4$, $|P"| \leq 3$, and the paths $P_n$ and $\widehat{P}_n$ are trivial.
• Figure 4: The proof of Proposition \ref{['prop:lg=qg']}.
• Figure 5: The proofs of Lemmas \ref{['lem:comp3']} and \ref{['lem:comp4']}. The red shaded area represents a copy of $\Gamma_{j,N}$ in $\Gamma(N)$.

Theorems & Definitions (58)

• Theorem 1.1: Hellyness of Relatively Hyperbolic Groups
• Theorem 1.2: Coarse Hellyness of Relatively Hyperbolic Groups
• Theorem 1.3: Parabolic Subgroups of (Coarsely) Helly Groups
• Theorem 1.4: Quasiconvex Subgroups of (Coarsely) Helly Groups
• Theorem 1.5: Geodesics to Quasi-geodesics
• Definition 2.1
• Lemma 2.2: see ghys-harpe
• Theorem 2.3: see ghys-harpe
• Remark 2.4
• Definition 2.5: Osin osin06