Arithmetic groups acting on compact manifolds.

*(English)*Zbl 0532.22012In the note the author announces some results concerning the volume preserving actions of arithmetic groups. In particular, the formulation and the indication of the proof of the following theorem is given.

Theorem A: Let G be a connected semisimple Lie group with a finite center, such that every simple factor of G has R-rank \(\geq 2\). Let \(\Gamma\subset G\) be a lattice. Let M be a compact Riemannian manifold, \(\dim M=n\). Set \(r=n^ 2+n+1\). Assume \(\Gamma\) acts by smooth isometries of M. Let \(\Gamma_ 0\subset \Gamma\) be a finite generating set. Then any volume preserving actions of \(\Gamma\) on M which (i) for elements of \(\Gamma_ 0\) is a sufficiently small \(C^ r\)-perturbation of the original action, and (ii) is ergodic, actually leaves a \(C^ 0\)- Riemannian metric invariant. In particular, there is a \(\Gamma\)-invariant topological distance function and the action is topologically conjugate to an action of \(\Gamma\) on a homogeneous space of a compact Lie group K defined via a dense range homomorphism of \(\Gamma\) into K.

The following conjecture is also formulated.

Conjecture: Let \(G,\Gamma\) be as above. Let \(d(G)\) be the minimal dimension of a nontrivial real representation of the Lie algebra of G, and \(n(G)\) be the minimal dimension of a simple factor of G. Let M be a compact manifold, \(\dim M=n>0\). Assume (i) \(n<d(G)\), and (ii) \(n(n+1)<2n(G)\). Then every volume preserving action on \(\Gamma\) on M is an action by a finite quotient of \(\Gamma\). In particular, there are no volume preserving ergodic actions of \(\Gamma\) on M.

Theorem A: Let G be a connected semisimple Lie group with a finite center, such that every simple factor of G has R-rank \(\geq 2\). Let \(\Gamma\subset G\) be a lattice. Let M be a compact Riemannian manifold, \(\dim M=n\). Set \(r=n^ 2+n+1\). Assume \(\Gamma\) acts by smooth isometries of M. Let \(\Gamma_ 0\subset \Gamma\) be a finite generating set. Then any volume preserving actions of \(\Gamma\) on M which (i) for elements of \(\Gamma_ 0\) is a sufficiently small \(C^ r\)-perturbation of the original action, and (ii) is ergodic, actually leaves a \(C^ 0\)- Riemannian metric invariant. In particular, there is a \(\Gamma\)-invariant topological distance function and the action is topologically conjugate to an action of \(\Gamma\) on a homogeneous space of a compact Lie group K defined via a dense range homomorphism of \(\Gamma\) into K.

The following conjecture is also formulated.

Conjecture: Let \(G,\Gamma\) be as above. Let \(d(G)\) be the minimal dimension of a nontrivial real representation of the Lie algebra of G, and \(n(G)\) be the minimal dimension of a simple factor of G. Let M be a compact manifold, \(\dim M=n>0\). Assume (i) \(n<d(G)\), and (ii) \(n(n+1)<2n(G)\). Then every volume preserving action on \(\Gamma\) on M is an action by a finite quotient of \(\Gamma\). In particular, there are no volume preserving ergodic actions of \(\Gamma\) on M.

Reviewer: G.A.Margulis

##### MSC:

22E40 | Discrete subgroups of Lie groups |

28D15 | General groups of measure-preserving transformations |

57S99 | Topological transformation groups |

20H99 | Other groups of matrices |

##### Keywords:

volume preserving actions; arithmetic subgroups; compact manifolds; rigidity; ergodic; finitely generated; invariant Riemannian metric; invariant topological distance function
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\textit{R. J. Zimmer}, Bull. Am. Math. Soc., New Ser. 8, 90--92 (1983; Zbl 0532.22012)

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##### References:

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