Abstract
We prove a Freiman–Ruzsa-type theorem valid in an arbitrary nilpotent group. Specifically, we show that a K-approximate group A in an s-step nilpotent group G is contained in a coset nilprogression of rank at most K O s ( 1 ) and cardinality at most exp ( K O s ( 1 ) ) | A | . To motivate this, we give a direct proof of Breuillard and Green's analogous result for torsion-free nilpotent groups, avoiding the use of Mal'cev's embedding theorem.
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