A graph is a 'shortcut graph' if there is a bound on the length of isometrically embedded cycles. This is a kind of combinatorial non-positive curvature. There is a coarsening of this property: a graph is 'strongly shortcut' if there is a bound on the length of $K$–biLipschitz cycles, for some $K > 1$. A group is strongly shortcut if it acts geometrically on some strongly shortcut graph. Examples include hyperbolic and CAT(0) groups.
Strongly shortcut groups are finitely presented and have polynomial Dehn function. We endeavor to show that these are not sufficient conditions by investigating a family of snowflake groups, which have known polynomial Dehn function. We show that these snowflake groups have the property that all of their asymptotic cones are simply connected, and at least some of these cones contain isometrically embedded circles. This implies that the Cayley graph of the defining presentation is not strongly shortcut, but leaves open the possibility that the groups admit some other action on a strongly shortcut graph. Separately, all previously known examples of groups with all of their asymptotic cones simply connected are of polynomial growth or have quadratic Dehn function or are certain combinations of such groups. Our examples are not in any of those classes. These are also the first examples that we know of groups whose asymptotic cones are simply connected but contain isometrically embedded circles.
This is joint work with Nima Hoda and Daniel Woodhouse.