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.

Figures (6)

• Figure 1: The closed outer path in black is a $1$-Lipchitz embedding $\alpha$ of a Riemannian circle $S$. This embedding $\alpha$ has poor bilipchitz constant but only because it badly distorts distances between relatively nearby pairs of points of $S$ (pairs contained in the subpaths $\alpha|_{Q_i}$). If we consider only antipodal pairs of points of $S$ then the distortion of their distances under $\alpha$ is much less than in these worst cases. In other words $\alpha$ has low distortion when viewed globally. The Circle Tightening Lemma tells us that if the global distortion of $\alpha$ is low enough then we can perform surgery on $\alpha$, replacing distorted subpaths of arbitrarily low total relative length with efficient alternatives (the red paths in the figure) in order to obtain an arbitrarily good bilipschitz constant.
• Figure 2: Continuing the pattern, one obtains an infinite graph that is strongly shortcut because it is the $1$-skeleton of a finite-dimensional $\mathop{\mathrm{CAT}}\nolimits(0)$ cube complex. Subdividing the interior edges of each $n \times n$ grid results in a quasi-isometric graph that is not strongly shortcut.
• Figure 3: In a circle tightening sequence, the circle $S_{i+1}$ is either equal to $S_i$ or is obtained from $S_i$ by replacing some geodesic segment $Q_i$ of $S_i$ with a shorter segment $\bar{Q}_i$, possibly of zero length.
• Figure 4: A circle tightening sequence that is disjoint up to $4$ but not disjoint up to $5$. The outer circle is the initial circle $S = S_0$. For $i \ge 0$, the circle $S_{i+1}$ is obtained from $S_i$ by replacing the geodesic segment $Q_i \subset S_{i}$ (indicated by perpendicular markings) with a shorter sequence $\bar{Q}_i$. The segment $Q_4$ (drawn in cyan) is the first replaced segment that cannot be viewed as a subspace of $S$ since it is not contained in $P^{\circ}_{0,4}$, which can be viewed as $S \setminus \bigcup_{i=0}^3 Q_i$.
• Figure 5: In the proof of Claim \ref{['claim:greedy_tightening_almost_disjoint']}, from a geodesic segment $Q$ of $S_j$ (indicated by a green outline) we obtain $Q^{-1}$ by removing the interiors of any $\bar{Q}_i$ that are partially contained in $Q$. We obtain $Q^{+} \to S_j$ by extending the inclusion $Q \hookrightarrow S_j$ to include full copies of any $\bar{Q}_i$ that are partially contained in $Q$. From $Q^{-} \hookrightarrow S_j$ and $Q^{+} \to S_j$ we obtain $Q_0^{-} \hookrightarrow S$ and $Q_0^{+} \to S$ in $S = S_0$ by replacing any $\bar{Q}_i \hookrightarrow S_j$ with $Q_i \hookrightarrow S$. The $\bar{Q}_i$ with $i < j$ are draw in red in $S_j$. The $Q_i$ with $i < j$ are drawn with perpendicular markings in $S$.

Theorems & Definitions (62)

• Theorem 1: Theorem \ref{['thm:not_sshortcut_equiv']}
• Theorem 2: Theorem \ref{['thm:graph_neg_equiv']}, Theorem \ref{['thm:not_sshortcut_equiv']}
• Theorem 3: Corollary \ref{['cor:ss_group']}
• Theorem 4: Theorem \ref{['thm:as_cat0_space']}, Theorem \ref{['thm:as_cat0_group']}
• Theorem 5: Corollary \ref{['cor:as_cones_ss']}
• Theorem 6: Corollary \ref{['cor:rough_approx_inv']}
• Theorem 7: Circle Tightening Lemma, Lemma \ref{['lem:circle_tightening']}
• Theorem 8: Fine Milnor-Schwarz Lemma, Lemma \ref{['lem:fine_ms']}, Remark \ref{['rmk:metric_proper_finite']}
• Remark 2.1
• Definition 2.2