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Dynamical systems and ergodic theory Definition

Zimmer program

Builds on Semisimple Lie groupLattice in a Lie groupSmooth manifoldDiffeomorphism

Definition

Let GG be a connected semisimple Lie group with finite center, all of whose simple factors have real rank at least 22. Let Γ\Gamma be a lattice in GG. Let MM be a compact smooth manifold and Diff(M)\mathrm{Diff}(M) its group of diffeomorphisms. The Zimmer program is the research program that aims to classify the homomorphisms ρ ⁣:Γ→Diff(M)\rho \colon \Gamma \to \mathrm{Diff}(M), with these expected outcomes:

  1. If dim⁡M\dim M is small compared with the rank of GG, then every such ρ\rho has finite image.
  2. If ρ\rho preserves a volume form, then, up to finite index, the action is built from isometric actions and from algebraic actions. Here an algebraic action is an action on a quotient Λ\H\Lambda \backslash H of a Lie group HH by a lattice Λ\Lambda. It comes from a homomorphism of Γ\Gamma or GG into HH.

Remarks

Example of an action in the sense of clause 2: SLn(ℤ)\mathrm{SL}_n(\Z) acts on the torus ℝn/ℤn\R^n / \Z^n by A⋅(x+ℤn)=Ax+ℤnA \cdot (x + \Z^n) = Ax + \Z^n. This is well defined because Aℤn=ℤnA \Z^n = \Z^n, and it preserves Lebesgue measure because det⁡A=1\det A = 1. Clause 1 is sharp for SLn(ℤ)\mathrm{SL}_n(\Z): it acts with infinite image on the real projective space of dimension n−1n - 1 through its linear action on lines of ℝn\R^n. The program is not a single theorem; clause 2 in particular has no fully agreed precise form. Precise test cases are Zimmer's conjecture for clause 1 and conjectures on actions preserving a rigid geometric structure for clause 2.

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Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptLattice in a Lie group, concept, no evidence levelLattice in a Lie groupconceptSemisimple Lie group, concept, no evidence levelSemisimple Lie groupconceptSmooth manifold, concept, no evidence levelSmooth manifoldconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptDiffeomorphism, concept, no evidence levelDiffeomorphismconceptZimmer program, definition, no evidence levelZimmer programdefinition
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. Lattice in a Lie group concept
  4. Semisimple Lie group concept
  5. Smooth manifold concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Differentiable function of several variables concept
  9. Diffeomorphism concept
  10. Zimmer program definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Zimmer's_conjecture
About this page
TagT5KZR54SG8
URIm2://T5KZR54SG8
Kinddefinition
WritingW0 stub
MSC37C85, 57S20, 22E40
DefinesZimmer program
UsesSemisimple Lie group, Lattice in a Lie group, Smooth manifold, Diffeomorphism
Changed2 November 2026

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