Directional asymptotic cones of groups equipped with bi-invariant metrics

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.