Strongly Shortcut Spaces

Abstract

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 CAT(0) 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 CAT(0) 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. Contents 1. Introduction 2 2. Basic notions and definitions 4 3. Characterizing the strong shortcut property 6 4. The Circle Tightening Lemma 12 5. The Fine Milnor-Schwarz Lemma 27 6. Asymptotically CAT(0) spaces 29 References 30

Asymptotic cones of snowflake groups and the strong shortcut property

“…The following theorem is a specialized form of a result of Hoda [25]. Theorem 6.0.1 (Circle Tightening Lemma [25,Theorem G]).…”

“…A graph Γ is strongly shortcut if, for some K > 1, there is a bound on the lengths of the K-biLipschitz cycles of Γ. This property has a natural generalization to rough geodesic metric spaces [25]. A group G is strongly shortcut if it satisfies one of the following three equivalent conditions [25]:…”

“…This property has a natural generalization to rough geodesic metric spaces [25]. A group G is strongly shortcut if it satisfies one of the following three equivalent conditions [25]:…”

“…There is a characterization of strongly shortcut spaces in terms of asymptotic cones [25] (see Section 2.3 for definitions): a space is strongly shortcut if and only if none of its asymptotic cones contains an isometrically embedded copy of a unit length circle with its length metric. One might wonder whether, for Cayley graphs of groups, this condition is equivalent to all asymptotic cones being simply connected.…”

“…The "fat" words are those such that the corresponding loop in Cayley graph is a K-biLipschitz loop for K close to 1. Thin words we cut into a bounded number of small pieces and at most one fat one using the "Circle Tightening Lemma" of [25]. This reduces the problem to constructing diagrams for K-biLipschitz loops for K arbitrarily close to 1.…”