This is joint work with Luc Guyot, Yves de Cornulier and Ralph Strebel.
The set G(m) of all isomorphism classes of m-generator groups can be identified with the set of all normal subgroups S of the free group F of rank m. G(m) is endowed with the Chabauty topology, which is based on the sets of all S containing one and avoiding a second given finite subset of F. It is a remarkable fact that whether a group G in G(m) is isolated or a condensation point, etc.… in this space is independent of its generating set and of m; hence those are group theoretic properties. I will present examples and condensation criteria, one of which is best understood in terms of the Geometric Invariant Σ(G) from joint work with Walter Neumann and Ralph Strebel.
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