We define the strong shortcut property for rough geodesic metric spaces,
generalizing the notion of strongly shortcut graphs. We show that the strong
shortcut property is a rough similarity invariant. We give several new
characterizations of the strong shortcut property, including an asymptotic cone
characterization. We use this characterization to prove that asymptotically
spaces are strongly shortcut. We prove 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. Thus we show
that
groups are strongly shortcut.
To prove these results, we use 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.
Keywords
strong shortcut property, asymptotically CAT(0) group,
nonpositively curved group, geometric group theory
