Strongly shortcut spaces (Q6632132)

Strongly shortcut graphs were introduced by the author [Trans. Am. Math. Soc. 375, No. 4, 2417--2458 (2022; Zbl 1520.20092)] as graphs satisfying a weak notion of non-positive curvature. The author here introduces the strong shortcut property for rough geodesic metric spaces, generalizing the notion of strongly shortcut graphs. He shows that the strong shortcut property is a rough similarity invariant. He gives several new characterizations of the strong shortcut property, including an asymptotic cone characterization and uses this characterization to prove that asymptotically CAT(0) spaces are strongly shortcut. Hence it is proved that if a group acts metrically properly and coboundedly on a strongly shortcut rough geodesic metric space, then it has a strongly shortcut Cayley graph and so is a strongly shortcut group. Therefore, the author shows that CAT(0) groups are strongly shortcut.\N\NIn order to prove these results the author uses several intermediate results which we believe may be of independent interest, including what we call the circle tightening lemma and the fine Milnor-Schwarz lemma. The circle tightening lemma describes how one may obtain a quasi-isometric embedding of a circle by performing surgery on a rough Lipschitz map from a circle that sends antipodal pairs of points far enough apart. The fine Milnor-Schwarz lemma is a refinement of the Milnor-Schwarz lemma that gives finer control on the multiplicative constant of the quasi-isometry from a group to a space it acts on.