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A relatively simple proof of the theorem was found by Bruce Kleiner. [5] Later, Terence Tao and Yehuda Shalom modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds. [6] [7] Gromov's theorem also follows from the classification of approximate groups obtained by Breuillard, Green ...
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 ...
Following ideas of Edward Witten, Gromov's work is also fundamental for Gromov-Witten theory, which is a widely studied topic reaching into string theory, algebraic geometry, and symplectic geometry. [39] [40] [41] From a different perspective, Gromov's work was also inspirational for much of Andreas Floer's work. [42]
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.
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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.