The finite images of finitely generated groups

Given any sequence of non-abelian finite simple primitive permutationgroups S_n, we construct a finitely generated group G whose profinitecompletion is the infinite permutational wreath product ...S_n S_n–1 ... S₀. It follows that the upper compositionfactors of G are exactly the groups S_n. By suitably choosingthe sequence S_n we can arrange that G has any one of a continuousrange of slow, non-polynomial subgroup growth types. We alsoconstruct a 61-generator perfect group that has every non-abelianfinite simple group as a quotient. 2000 Mathematics SubjectClassification: 20E07, 20E08, 20E18, 20E32.