  We consider the oriented graph whose vertices are isomorphism classes of
finitely generated groups, with an edge from G to H if, for some generating set
T in H and some sequence of generating sets S_i in G, the marked balls of
radius i in (G,S_i) and in (H,T) coincide.
  Given a nilpotent group G, we characterize its connected component in this
graph: if that connected component contains at least one torsion-free group,
then it consists of those groups which generate the same variety of groups as
G.
  The arrows in the graph define a preorder on the set of isomorphism classes
of finitely generated groups. We show that a partial order can be imbedded in
this preorder if and only if it is realizable by subsets of a countable set
under inclusion.
  We show that every countable group imbeds in a group of non-uniform
exponential growth. In particular, there exist groups of non-uniform
exponential growth that are not residually of subexponential growth and do not
admit a uniform imbedding into Hilbert space.
