Year: 2015
Type: article
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.
Source: Annales de l’institut Fourier
DOI: 10.5802/aif.2984
Authors: Laurent Bartholdi, Anna Erschler
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: bronze
OpenAlex ID: W2208572199