Strong shortcuts, generating sets, and isometric circles in asymptotic cones

TL;DR

This paper investigates how the choice of generating sets affects the existence of shortcuts in Cayley graphs and demonstrates that certain groups can have asymptotic cones with embedded circles that are null-homotopic.

Contribution

It shows that the ability to shortcut loops in Cayley graphs depends on the generating set and provides examples with asymptotic cones containing null-homotopic embedded circles.

Findings

Loop shortcuts depend on generating set choice

Example of a group with asymptotic cones containing null-homotopic circles

Demonstrates variability in geometric properties of asymptotic cones

Abstract

We show that whether loops can be shortcut in a group's Cayley graph depends on the choice of finite generating set. Our example is the direct product of two rank-2 free groups and a consequence is that this group has asymptotic cones with isometrically embedded circles that are null-homotopic.