Shortcut Graphs and Groups

Abstract

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

Strongly Shortcut Spaces

“…A graph Γ is strongly shortcut if, for some K > 1, there is a bound on the lengths of cycles α : S → Γ for which d X α(p), α(p) ≥ 1 K • |S| 2 for every antipodal pair of points p, p ∈ S. By a theorem of this author, a graph Γ is strongly shortcut if and only if, for some K > 1, there is a bound on the lengths of the K-bilipschitz embedded cycles of Γ [4].…”

“…Strongly shortcut graphs were introduced in earlier work of this author [4] as graphs satisfying a weak notion of nonpositive curvature. They were shown to unify a broad family of graphs of interest in geometric group theory and metric graph theory, including hyperbolic graphs, 1-skeleta of finite dimensional CAT(0) cube complexes and standard Cayley graphs of finitely generated Coxeter groups [4]. Strongly shortcut groups are defined as those groups admitting a proper and cocompact action on a strongly shortcut graph [4].…”

“…They were shown to unify a broad family of graphs of interest in geometric group theory and metric graph theory, including hyperbolic graphs, 1-skeleta of finite dimensional CAT(0) cube complexes and standard Cayley graphs of finitely generated Coxeter groups [4]. Strongly shortcut groups are defined as those groups admitting a proper and cocompact action on a strongly shortcut graph [4]. They are finitely presented and have polynomial isoperimetric functions and so have decidable word problem [4].…”

“…Strongly shortcut groups are defined as those groups admitting a proper and cocompact action on a strongly shortcut graph [4]. They are finitely presented and have polynomial isoperimetric functions and so have decidable word problem [4].…”

Relatively hyperbolic groups with strongly shortcut parabolics are strongly shortcut

“…Strongly shortcut graphs and groups were introduced by the first named author [Hod18] who later generalized the strong shortcut property to rough geodesic metric spaces [Hod20]. The strong shortcut property is a very general form of nonpositive curvature condition satisfied by many spaces of interest in geometric group theory, metric graph theory and geometric topology.…”