Abstract
Let L m be a free group on m fixed generators. We study properties of the space G m of all the (finitely generated) quotient groups of L m . This space has a natural topology, in which it is compact, and a natural equivalence relation, the relation of isomorphism. We prove that the quotient of the borelian structure of G m by this equivalence relation is not standard. Furthermore, we show, using methods inspired from those of Gromov, that some “exotic” groups are generic in the closure, in G m , of non-elementary Gromov-hyperbolic groups without torsion, or more generally with cyclic centralizers.
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