Ordering the space of finitely generated groups

TL;DR

This paper introduces a graph structure on finitely generated groups based on their local geometric properties, characterizes components for nilpotent groups, and explores the embedding of partial orders and groups of non-uniform exponential growth.

Contribution

It defines a new graph-based ordering of finitely generated groups, characterizes components for nilpotent groups, and demonstrates embeddings of partial orders and groups with non-uniform exponential growth.

Findings

Connected components for nilpotent groups relate to group varieties.

Partial orders can be embedded if realizable by subsets of a countable set.

Existence of groups with non-uniform exponential growth not residually subexponential.

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 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.