The Morse Local-to-Global Property for Graph Products

TL;DR

The paper proves that graph products of infinite Morse local-to-global groups are themselves Morse local-to-global. It develops a maximization framework for relative hierarchically hyperbolic spaces (relative HHS) under mild hypotheses to produce a maximized structure whose top-level space is hyperbolic and captures Morse quasi-geodesics, yielding Morse detectability. It then links stable embeddings to projections onto the top-level space, establishing a stable-embedding criterion and applying it to graph products with no isolated vertices. The results provide a robust method to certify the Morse local-to-global property for a broad class of graph products and identify stable subgroups via orbit maps into the top-level hyperbolic space, with broad implications for structure and geometry of these groups.

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.

Figures (3)

• Figure 1: The procedure of building nesting chains as described in the large link axiom. All black lines represent set containment (identifying the $T^p_q$ with $\{T^p_q\}$). The red lines represent applying the large link axiom to $T^p_q$ domain at the top, and as such every individual domain in the $A^{p+1}_q$ below nests into the $T^p_q$ above it.
• Figure 2: A depiction of $CS$ in the proof of condition (3) that $\gamma(\mathcal{Y})$ is $D'$-contracting.
• Figure 3: A depiction of $\mathcal{X}$ in the proof that $\gamma(\mathcal{Y})$ has $D'$-bounded projections.

Theorems & Definitions (95)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Proposition 2.5: gromov1987hyperbolic
• Definition 2.6
• Definition 2.7