Weak Morse properties in spaces with bounded combings

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.

30 pages (20 paper, 10 appendix), 8 figures