Asymptotic cones of snowflake groups and the strong shortcut property

TL;DR

This paper constructs an infinite family of snowflake groups with simply connected asymptotic cones, exhibiting novel geometric properties where asymptotic cones contain nontrivial loops without topologically nontrivial ones.

Contribution

It introduces new examples of groups with simply connected asymptotic cones that contain isometrically embedded circles, challenging previous assumptions about group geometry.

Findings

All groups have simply connected asymptotic cones.

Existence of asymptotic cones with embedded circles.

Groups are neither polynomial growth nor quadratic Dehn function.

Abstract

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.