Asymptotic cones of snowflake groups and the strong shortcut property

TL;DR

This paper constructs an infinite family of snowflake groups $G_L$ (even $L" lat 6$) that are not virtually nilpotent and have superquadratic Dehn functions, yet all asymptotic cones are simply connected. The authors establish a Loop Subdivision Criterion linking cone simple connectedness to uniformly bounded-area fillings of boundary words, and then develop a detailed fat/skinny analysis of biLipschitz loops. By analyzing geodesics, line intersections, and distortion of powers of $a$, $x$, and $y$, they implement an inductive filling procedure for $K$-biLipschitz loops, decomposing diagrams into central regions, enfilades, and branching regions, and filling each piece with carefully bounded area/mesh. The results yield that all asymptotic cones of $G_L$ are simply connected, while some asymptotic cone contains an isometrically embedded circle, showing that simple connectivity does not imply the strong shortcut property for Cayley graphs. The work thus provides the first counterexamples separating asymptotic-cone simple connectivity from strong shortcut behavior in groups and raises open questions about the generating-set dependence of strong shortcut and related coarse-geometric phenomena.

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.

Figures (13)

• Figure 1: A snowflake for $a^{L^7}$.
• Figure 2: $|a^\ell|$ versus $\ell$ for $L=10$. Local minima occur when $L|\ell$, second order minima occur when $L|\ell^2$, etc.
• Figure 3: Subdivided snowflake to depth 1.
• Figure 4: Capping off a branch
• Figure 5: biLipschitz snowflake

Theorems & Definitions (104)

• Theorem : Theorem \ref{['maintheorem']}
• Theorem : Corollary \ref{['cor:isometriccircle']}
• Definition 2.2.1
• Definition 2.3.1
• Theorem 2.3.2: Loop Subdivision Property brady2007geometry
• Remark 2.3.3
• Proposition 3.1.1
• Remark 3.1.2
• proof
• Lemma 3.2.1