Math 2.0

Dynamical systems and ergodic theory Conjecture

Zimmer's conjecture

Builds on Lattice in a Lie groupSpecial linear groupSmooth manifoldDiffeomorphism

Statement

Let n≥3n \ge 3 and let Γ\Gamma be a lattice in the special linear group SLn(ℝ)\mathrm{SL}_n(\R), for example Γ=SLn(ℤ)\Gamma = \mathrm{SL}_n(\Z). Let MM be a compact smooth manifold with dim⁡M<n−1\dim M < n - 1, and let Diff(M)\mathrm{Diff}(M) be its group of diffeomorphisms. Then every group homomorphism ρ ⁣:Γ→Diff(M)\rho \colon \Gamma \to \mathrm{Diff}(M) has finite image.

Remarks

This is the case of SLn(ℝ)\mathrm{SL}_n(\R) of a more general conjecture of Zimmer for lattices in higher-rank simple Lie groups, with a dimension bound depending on the group. Zimmer stated conjectures of this kind in his 1986 address to the International Congress of Mathematicians, in several versions, some of them only for actions preserving a volume form. The bound n−1n - 1 is sharp: SLn(ℤ)\mathrm{SL}_n(\Z) acts with infinite image on the real projective space of dimension n−1n - 1. Aaron Brown, David Fisher and Sebastián Hurtado proved it, with C2C^2 regularity, for cocompact lattices in SLn(ℝ)\mathrm{SL}_n(\R). In a paper published in Inventiones Mathematicae in 2020 they proved it for actions by C2C^2 diffeomorphisms of finite-index subgroups of SLn(ℤ)\mathrm{SL}_n(\Z) with n>3n > 3. The general conjecture for lattices in other higher-rank groups is still studied.

Reading path

To understand this page from the ground up, read these 19 pages in order. The foundations come first.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptGeneral linear group, object, no evidence levelGeneral linear groupobjectIdentity matrix, object, no evidence levelIdentity matrixobjectLattice in a Lie group, concept, no evidence levelLattice in a Lie groupconceptMatrix, definition, no evidence levelMatrixdefinitionSmooth manifold, concept, no evidence levelSmooth manifoldconceptVector space, definition, no evidence levelVector spacedefinitionTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptSquare matrix, definition, no evidence levelSquare matrixdefinitionLinear map, definition, no evidence levelLinear mapdefinitionDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptMultilinear map, definition, no evidence levelMultilinear mapdefinitionDiffeomorphism, concept, no evidence levelDiffeomorphismconceptAlternating multilinear map, definition, no evidence levelAlternating multiline…definitionDeterminant, definition, no evidence levelDeterminantdefinitionSpecial linear group, object, no evidence levelSpecial linear groupobjectZimmer's conjecture, conjecture, no evidence levelZimmer's conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. General linear group object
  4. Identity matrix object
  5. Lattice in a Lie group concept
  6. Matrix definition
  7. Smooth manifold concept
  8. Vector space definition
  9. Total derivative concept
  10. Function of several variables concept
  11. Square matrix definition
  12. Linear map definition
  13. Differentiable function of several variables concept
  14. Multilinear map definition
  15. Diffeomorphism concept
  16. Alternating multilinear map definition
  17. Determinant definition
  18. Special linear group object
  19. Zimmer's conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Zimmer's_conjecture
About this page
TagJW71J3AKY4
URIm2://JW71J3AKY4
Kindconjecture
WritingW0 stub
MSC37C85, 57S20, 22E40
StatusThis copy of the site has no build data. Run m2 status JW71J3AKY4.
AboutZimmer program
UsesLattice in a Lie group, Special linear group, Smooth manifold, Diffeomorphism

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