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There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of Joseph A. Wolf [2] showed that if G is a finitely generated nilpotent group, then the group has polynomial growth. Yves Guivarc'h [3] and independently Hyman Bass [4] (with different proofs
The free abelian group has a polynomial growth rate of order d. The discrete Heisenberg group has a polynomial growth rate of order 4. This fact is a special case of the general theorem of Hyman Bass and Yves Guivarch that is discussed in the article on Gromov's theorem.
See Gromov's theorem on groups of polynomial growth. (Also see D. Edwards for an earlier work.) The key ingredient in the proof was the observation that for the Cayley graph of a group with polynomial growth a sequence of rescalings converges in the pointed Gromov–Hausdorff sense.
One of Gromov's compactness theorems: Gromov's compactness theorem (geometry) in Riemannian geometry; Gromov's compactness theorem (topology) in symplectic topology; Gromov's Betti number theorem Gromov–Ruh theorem on almost flat manifolds; Gromov's non-squeezing theorem in symplectic geometry; Gromov's theorem on groups of polynomial growth
Theorems which use quasi-isometry invariants to prove algebraic results about groups, for example: Gromov's polynomial growth theorem; Stallings' ends theorem; Mostow rigidity theorem. Quasi-isometric rigidity theorems, in which one classifies algebraically all groups that are quasi-isometric to some given group or metric space.
The Tits alternative is an important ingredient [2] in the proof of Gromov's theorem on groups of polynomial growth. In fact the alternative essentially establishes the result for linear groups (it reduces it to the case of solvable groups, which can be dealt with by elementary means).
An important result in the subject is Gromov's theorem on groups of polynomial growth, obtained by Gromov in 1981, which shows that a finitely generated group has polynomial growth if and only if this group has a nilpotent subgroup of finite index.
Gromov's theorem on groups of polynomial growth; N. Nilpotent group; Nilsequence This page was last edited on 15 November 2014, at 20:41 (UTC). Text is available ...