  The objective of this series is to study metric geometric properties of
(coarse) disjoint unions of amenable Cayley graphs. We employ the Cayley
topology and observe connections between large scale structure of metric spaces
and group properties of Cayley accumulation points. In this Part I, we prove
that a disjoint union has property A of G. Yu if and only if all groups
appearing as Cayley accumulation points in the space of marked groups are
amenable. As an application, we construct two disjoint unions of finite special
linear groups (and unimodular linear groups) with respect to two systems of
generators that look similar such that one has property A and the other does
not admit (fibred) coarse embeddings into any Banach space with non-trivial
type (for instance, any uniformly convex Banach space).
