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Ordering the space of finitely generated groups. (Comment ordonner l’espace des groupes de type fini.) (English. French summary) Zbl 1372.20030

Summary: 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.

MSC:

20E10 Quasivarieties and varieties of groups
20E34 General structure theorems for groups
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
20F69 Asymptotic properties of groups
20E22 Extensions, wreath products, and other compositions of groups
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