Iterated asymptotic cones

TL;DR

This paper explores the diverse behaviors of iterated asymptotic cones in metric spaces and groups, demonstrating the existence of groups with highly varied asymptotic cone structures, including non-homeomorphic and periodically homeomorphic cones.

Contribution

It introduces a new class of metric spaces with varied iterated asymptotic cone behaviors and constructs groups exhibiting these phenomena, expanding understanding of asymptotic cone diversity.

Findings

Existence of groups with pairwise non-homeomorphic iterated cones

Existence of groups with periodically homeomorphic iterated cones

Demonstration of wide-ranging behaviors in iterated asymptotic cones

Abstract

Iterated asymptotic cones have been used by Dru\c{t}u and Sapir to construct a group with uncountably many pairwise non-homeomorphic asymptotic cones. In this paper we define a class of metric spaces which display a wide range of behaviors with respect to iterated asymptotic cones, and we use those to construct examples within the class of groups. Namely, we will show that there exists a group whose iterated cones are pairwise non-homeomorphic, or periodically homeomorphic.