Asymptotic cones of snowflake groups and the strong shortcut property

We exhibit an infinite family of snowflake groups all of whose asymptotic cones are simply connected.Our groups have neither polynomial growth nor quadratic Dehn function, the two usual sources of this phenomenon.We further show that each of our groups has an asymptotic cone containing an isometrically embedded circle or, equivalently, has a Cayley graph that is not strongly shortcut.These are the first examples of groups whose asymptotic cones contain "metrically nontrivial" loops but no topologically nontrivial ones.20F65, 20F69, 51F30 1. Introduction 3429 2. Preliminaries 3433 3. Snowflake groups 3436 4. The snowflake Cayley graph is not strongly shortcut 3448 5.The idea of the proof that asymptotic cones are simply connected 3451 6. Reduction to bi-Lipschitz cycles 3455 7. Bi-Lipschitz cycles in snowflake groups 3457 8. Constructing fillings 3483 9. Proof of the main theorem and closing remarks 3498 References 3499