Ordering the space of finitely generated groups

Abstract

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 (H,T) coincide. We show that if a connected component of this graph contains at least one torsion-free nilpotent group G, then it consists of those groups which generate the same variety of groups as G. We show on the other hand that the first Grigorchuk group has infinite girth, and hence belongs to the same connected component as free groups. 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.