| Gromov's Theorem On Groups Of Polynomial Growth |
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Information AboutGromov's Theorem On Groups Of Polynomial Growth |
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Groups of ''polynomial'' growth, as those groups which have Nilpotent subgroups of finite Index . The Growth Rate of a group is a Well-defined notion from Asymptotic Analysis . To say that a finitely generated group has polynomial growth means the number of elements of Length (relative to a symmetric generating set) at most ''n'' is bounded above by a Polynomial function ''p''(''n''). The ''order of growth'' is then the Degree of the polynomial function ''p''. A ''nilpotent'' group ''G'' is a group with a Lower Central Series terminating in the identity subgroup. Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index. There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of Joseph A. Wolf showed that if ''G'' is a finitely generated nilpotent group, then the group has polynomial growth. Hyman Bass computed the exact order of polynomial growth. Let ''G'' be a finitely generated nilpotent group with lower central series : In particular, the quotient group ''Gk/Gk+1'' is a finitely generated abelian group. Bass's theorem states that the order of polynomial growth of ''G'' is : where: rank In particular, Gromov's and Bass's theorems imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding for example, fractional powers). In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called the Gromov-Hausdorff Convergence , is currently widely used in geometry. REFERENCES
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