Local topological properties of asymptotic cones of groups

TL;DR

This paper introduces a local property related to Gromov's loop division to analyze the topological structure of asymptotic cones of groups, providing conditions for uncountable fundamental groups and insights into their local topology.

Contribution

It defines a new local property for asymptotic cones and demonstrates its use in characterizing their fundamental groups and local topological features.

Findings

Identifies conditions for asymptotic cones to have uncountable fundamental groups

Provides a framework to analyze local topological structures of asymptotic cones

Applies the property to various groups in the literature

Abstract

We define a local analogue to Gromov's loop division property which is use to give a sufficient condition for an asymptotic cone of a complete geodesic metric space to have uncountable fundamental group. As well, this property is used to understand the local topological structure of asymptotic cones of many groups currently in the literature.