Directional asymptotic cones of groups equipped with bi-invariant metrics

TL;DR

This paper introduces a new construction called the directional asymptotic cone for groups with bi-invariant metrics, avoiding ultrafilters and providing a canonical Lipschitz homomorphism to the standard asymptotic cone.

Contribution

It presents a novel, ultrafilter-free method to construct asymptotic cones, resulting in a contractible topological group with desirable metric properties.

Findings

The directional asymptotic cone is a contractible topological group.

It admits a canonical Lipschitz homomorphism to the standard asymptotic cone.

The cone of a countable group is separable.

Abstract

Given a bi-invariant metric on a group, we construct a version of an asymptotic cone without using ultrafilters. The new construction, called the directional asymptotic cone, is a contractible topological group equipped with a complete bi-invariant metric and admits a canonical Lipschitz homomorphism to the standard asymptotic cone. Moreover, the directional asymptotic cone of a countable group is separable.