Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut

TL;DR

This paper proves that groups hyperbolic relative to strongly shortcut groups are themselves strongly shortcut, providing new examples and establishing a key action property of such groups on specific graphs.

Contribution

It demonstrates that relative hyperbolicity with strongly shortcut parabolics implies the entire group is strongly shortcut, introducing a new class of examples.

Findings

Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut.

Every such group acts properly and cocompactly on a graph with convex subgraphs for parabolics.

New examples of strongly shortcut groups are identified.

Abstract

We show that a group that is hyperbolic relative to strongly shortcut groups is itself strongly shortcut, thus obtaining new examples of strongly shortcut groups. The proof relies on a result of independent interest: we show that every relatively hyperbolic group acts properly and cocompactly on a graph in which the parabolic subgroups act properly and cocompactly on convex subgraphs.