Weak Morse properties in spaces with bounded combings

TL;DR

This work links two coarse geometric notions of non‑positive curvature—bounded combings and the Morse local‑to‑global (MLTG) property—and proves that a geodesic space with a bounded quasi‑geodesic combing and a $\sigma$‑compact Morse boundary satisfies the MLTG property. It develops a globalization framework for quasi‑geodesics, proves that bounded combings yield the weak MLTG, and shows that sigma‑compactness of the Morse boundary upgrades this to the strong MLTG. A key dichotomy is established: any space with the weak MLTG and a cobounded isometry action either has the full MLTG or a non‑$\sigma$‑compact Morse boundary, with the reverse direction tied to known results asserting that MLTG implies $\sigma$‑compact Morse boundary. The results unify and extend the Morse boundary program, yielding algorithms and language‑theoretic consequences, stable subgroups theory, and growth‑gap phenomena in wide classes of spaces and groups.

Abstract

We relate two notions of non-positive curvature: bounded combings and the Morse local-to-global (MLTG) property (in its weak and strong version). The latter is a property of a space that has been shown to eliminate pathological behavior of Morse geodesics. We showcase its importance in a survey in the appendix. We show that having a bounded combing implies the weak MLTG property. If the Morse boundary of a group is sigma-compact, we show that the weak MLTG property is upgraded to the (strong) MLTG property.

Figures (8)

• Figure 1: Proof of Lemma \ref{['lem:reverse_inclusion_QG_nbhd']}
• Figure 2: Proof of Lemma \ref{['lemma:quasi-geodesics_stay']}. Dotted lines have length at most $D$ and every point on the blue segment has distance at most $\epsilon(D)$ from $x_2$.
• Figure 3: The point $y = \eta(t)$ is a $(D, \ell)$--exit point.
• Figure 4: Dependency of constants in the proof of Theorem \ref{['thm:mltg']}. An arrow from $x$ to $y$ implies that $y$ depends on $x$.
• Figure 5: Setup for proof of Theorem \ref{['thm:mltg']}

Theorems & Definitions (84)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Weak Morse local-to-global
• Theorem 1.5
• Corollary 1.6
• Corollary 1.7: RussellSprianoTran:thelocal and Lemma \ref{['appendix:improved_malnormal']}, abundance of free subgroups
• Theorem 1.8: CordesRussellSprianoZalloum:regularity and Lemma \ref{['appendix:improved_malnormal']}, growth gap
• Definition 2.2: Quasi-isometric embedding, quasi-isometry, quasi-geodesic
• Lemma 2.4: Improved quasi-geodesics BridsonHaefliger, Lemma 1.11, Burago-Ivanov, Proposition 8.3.4
• Lemma 2.6: Containment implies finite Hausdorff distance