Finitely presented groups with infinitely many non-homeomorphic asymptotic cones

TL;DR

This paper constructs a finitely presented group with infinitely many non-homeomorphic asymptotic cones and demonstrates that the presence of cut points in these cones depends on the choice of scaling constants and ultrafilters.

Contribution

It introduces a specific finitely presented group with infinitely many non-homeomorphic asymptotic cones and explores the dependence of cut points on ultrafilter choices.

Findings

Existence of a finitely presented group with infinitely many non-homeomorphic asymptotic cones

Cut points in asymptotic cones depend on ultrafilter and scaling constants

Dependence of asymptotic cone properties on ultrafilter choice

Abstract

We construct a finitely presented group with infinitely many non-homeomorphic asymptotic cones. We also show that the existence of cut points in asymptotic cones of finitely presented groups does, in general, depend on the choice of scaling constants and ultrafilters.