Strong shortcuts, generating sets, and isometric circles in 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.

arXiv comment6 pages, 2 figures