L’espace des groupes de type fini

Let Lm be a free group on m fixed generators. We study properties of the space Gm of all the (finitely generated) quotient groups of Lm. 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 Gm 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 Gm, of non-elementary Gromov-hyperbolic groups without torsion, or more generally with cyclic centralizers.