Ultrafilters and the asymptotic cones Con∞X and ConωX; connectivity of Con∞ for Lie groups and lattices; geometry of Con∞ and the space of the word metrics.

The invariants of X in the previous §1 were produced by performing standard topological constructions on the large scale. Here we want to proceed differently by defining a canonical asymptotic cone of X at ∞ whose ordinary topological invariants will serve as large scale invariants of X. This cone Con∞X captures the geometry rather than the topology of X on the large scale and it can be described informally as follows. Let us imagine an observer who moves away from a metric space X and from time to time makes an observation consisting in measuring finitely many distances between certain points in X. When the observer is located at distance d away from X he concentrates on the distances which appear to him of bounded magnitude (as he has a limited field of vision) which correspond to distances proportional to d on the real scale. As the observer goes further away at some distance d' which is much greater than d, the points seen earlier from the distance d become indistinguishable; but now he is concerned with points within distance about d' in X. It may happen by pure accident that the result of the d'–observation is similar to the d–observation despite the fact that in reality (i.e. in X) the observed objects have nothing in common.