---
tag: BRKPE99MFC
kind: concept
name: Zimmer program
msc: ["37C85", "57S20", "22E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Zimmer's_conjecture
---

## Summary

The Zimmer program is the study of smooth actions of [[lattices|BTYF5Y5PFY]] in higher-rank [[semisimple Lie groups|XJS07AJ50Y]] on compact manifolds.
Its guiding idea is that such actions are very restricted.
In low dimension they should be finite, and in general they should come from algebraic constructions.
It extends the [[Margulis superrigidity theorem|DSKV65NHZZ]], which describes linear representations of these lattices, to actions by diffeomorphisms.
[[Zimmer's conjecture|JW71J3AKY4]] is its best known test case.

## Explanations

### L1 human

A group such as $\mathrm{SL}_3(\Z)$ is huge and rigid.
The Zimmer program asks in which ways such a group can act by smooth symmetries on a compact space like a circle, a sphere or a torus.
The expected answer is that it can hardly act at all on small spaces, and that on larger spaces its actions come from known algebraic models.
For example, $\mathrm{SL}_3(\Z)$ acts on the torus $\R^3/\Z^3$ through its linear action on $\R^3$, and actions of this kind are expected to be typical.