Shortcut graphs and groups

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 generalised 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.…”

“…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. These include Gromov-hyperbolic spaces [Hod18], asymptotically CAT(0) spaces [Hod20], hierarchically hyperbolic spaces, coarse Helly metric spaces of uniformly bounded geometry [HHP20], 1-skeletons of finite dimensional CAT(0) cube complexes (i.e. median graphs), 1-skeletons of quadric complexes (i.e.…”

“…bridged graphs), standard Cayley graphs of Coxeter groups [Hod18] and all of the Thurston geometries except Sol [HP,Kar11]. Despite this surprisingly unifying nature, there are nonetheless important consequences for groups that act metrically properly and coboundedly on strongly shortcut geodesic metric spaces: finite presentability, polynomial isoperimetric function and thus decidable word problem [Hod18,Hod20].…”

Asymptotic cones of snowflake groups and the strong shortcut property

“…The strong shortcut property was introduced by the second named author in order to explore commonalities between the many theories of nonpositively curved groups which have been developed in recent decades [23]. 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.…”

“…Gromov-hyperbolic spaces [18] and SL(2, R) with the Sasaki metric [28]. • 1-skeletons of systolic [40,15,22,27], quadric [3,24] and finite dimensional CAT(0) cubical [2,32,18] complexes [23]. • Standard Cayley graphs of Coxeter groups [23,33].…”