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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) computed the exact order of polynomial growth.
The non-squeezing theorem, also called Gromov's non-squeezing theorem, is one of the most important theorems in symplectic geometry. [1] It was first proven in 1985 by Mikhail Gromov. [2] The theorem states that one cannot embed a ball into a cylinder via a symplectic map unless the radius of the ball is less than or equal to the radius of the ...
Gromov's theorem may mean one of a number of results of Mikhail Gromov: 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
The role of this theorem in the theory of Gromov–Hausdorff convergence may be considered as analogous to the role of the Arzelà–Ascoli theorem in the theory of uniform convergence. [2] Gromov first formally introduced it in his 1981 resolution of the Milnor–Wolf conjecture in the field of geometric group theory, where he applied it to ...
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).
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem , and is the key point in the proof of Gromov's compactness theorem .
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. The lamplighter group has an exponential growth. The existence of groups with intermediate growth, i.e. subexponential but not polynomial was open for many years.
Gromov's almost flat manifolds. Astérisque, 81. Société Mathématique de France, Paris, 1981. 148 pp. Peter Buser and Hermann Karcher. The Bieberbach case in Gromov's almost flat manifold theorem. Global differential geometry and global analysis (Berlin, 1979), pp. 82–93, Lecture Notes in Math., 838, Springer, Berlin-New York, 1981.
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