The Morse Local-to-Global Property for Graph Products

TL;DR

This paper extends the Morse local-to-global property to graph products of infinite groups, showing they retain this property under certain conditions, and characterizes stable embeddings in hierarchically hyperbolic spaces.

Contribution

It generalizes the Morse local-to-global property to graph products and characterizes stable embeddings in hierarchically hyperbolic spaces.

Findings

Graph products of infinite Morse local-to-global groups have the Morse local-to-global property.

Stable embeddings are quasi-isometric embeddings into the top hyperbolic space.

Graph products of groups with no isolated vertices are Morse detectable.

Abstract

The Morse local-to-global property generalizes the local-to-global property for quasi-geodesics in a hyperbolic space. We show that graph products of infinite Morse local-to-global groups have the Morse local-to-global property. To achieve this, we generalize the maximization procedure of Abbott, Behrstock, and Durham for relatively hierarchically hyperbolic groups with clean containers. Under mild conditions satisfied by graph products, we show that stable embeddings into a relatively hierarchically hyperbolic space are exactly those which are quasi-isometrically embedded in the top level hyperbolic space by the orbit map. This shows that graph products of any infinite groups with no isolated vertices are Morse detectable.