Abstract
Let F be a relatively free algebra of infinite rank ϰ. We say that F has the small index property if any subgroup of Γ = Aut(F) of index at most ϰ contains the pointwise stabilizer Γ(U) of a subset U of F of cardinality less than ϰ. We prove that every infinitely generated free nilpotent/abelian group has the small index property, and discuss a number of applications.
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