Gromov's theorem on groups of polynomial growth states that any finitely generated group with polynomial growth is virtually nilpotent. This theorem is a significant result in geometric group theory and has important implications for the structure of groups.
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Every finitely generated group of polynomial growth of a group has a virtually nilpotent group structure. Applying the one-scale virtual nilpotence theorem gives a finite normal subgroup with nilpotent quotient; centralizing that finite subgroup produces a nilpotent subgroup of finite index.