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The order of growth is then the least degree of any such 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.
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.
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
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).
For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups. This terminology is also used when P is just another group.
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.
The Breuillard–Green–Tao theorem on classification of approximate groups can be used to give a new proof of Gromov's theorem on groups of polynomial growth.The result obtained is actually a bit stronger since it establishes that there exists a "growth gap" between virtually nilpotent groups (of polynomial growth) and other groups; that is, there exists a (superpolynomial) function such ...
Grinberg's theorem (graph theory) Gromov's compactness theorem (Riemannian geometry) Gromov's compactness theorem (symplectic topology) Gromov's theorem on groups of polynomial growth (geometric group theory) Gromov–Ruh theorem (differential geometry) Gross–Zagier theorem (number theory)