We relate three classes of nonpositively curved metric spaces: hierarchically hyperbolic spaces, coarsely Helly spaces, and strongly shortcut spaces. We show that any hierarchically hyperbolic space admits a new metric that is coarsely Helly. The new metric is quasi-isometric to the original metric and is preserved under automorphisms of the hierarchically hyperbolic space. We show that any coarsely Helly metric space of uniformly bounded geometry is strongly shortcut. Consequently, hierarchically hyperbolic groups-including mapping class groups of surfaces-are coarsely Helly and coarsely Helly groups are strongly shortcut.Using these results we deduce several important properties of hierarchically hyperbolic groups, including that they are semihyperbolic, have solvable conjugacy problem, are of type F P8, have finitely many conjugacy classes of finite subgroups, and their finitely generated abelian subgroups are undistorted. Along the way we show that hierarchically quasiconvex subgroups of hierarchically hyperbolic groups have bounded packing.
We introduce shortcut graphs and groups. Shortcut graphs are graphs in which cycles cannot embed without metric distortion. Shortcut groups are groups which act properly and cocompactly on shortcut graphs. These notions unify a surprisingly broad family of graphs and groups of interest in geometric group theory and metric graph theory, including: the 1-skeleta of systolic and quadric complexes (in particular finitely presented C(6) and C(4)-T(4) small cancellation groups), 1-skeleta of finite dimensional CAT(0) cube complexes, hyperbolic graphs, standard Cayley graphs of finitely generated Coxeter groups and the standard Cayley graph of the Baumslag-Solitar group BS(1, 2). Most of these examples satisfy a strong form of the shortcut property.The shortcut properties also have important geometric group theoretic consequences. We show that shortcut groups are finitely presented and have exponential isoperimetric and isodiametric functions. We show that groups satisfying the strong form of the shortcut property have polynomial isoperimetric and isodiametric functions. Contents 1. Introduction 1 2. Definitions 3 3. Basic properties 4 4. Filling properties and disk diagrams 11 5. Combinations 13 6. Examples 18 7. The Baumslag-Solitar group BS(1, 2) 25 References 35
We prove that asymptotic cones of Helly groups are countably hyperconvex. We use this to show that virtually nilpotent Helly groups are virtually abelian and to characterize virtually abelian Helly groups via their point groups. We apply this to prove that the 3-3-3-Coxeter group is not Helly, thus obtaining the first example of a systolic group that is not Helly, answering a question of Chalopin, Chepoi, Genevois, Hirai, and Osajda.
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
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