Real-analytic, volume-preserving actions of lattices on 4-manifolds

Benson Farb and Peter Shalen ^{∗}

January 17, 2002

Abstract

We prove that any real-analytic, volume-preserving action of a lattice Γ in a simple Lie group with Q-rank(Γ) ≥ 7 on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.

1 Results

Zimmer conjectured in [Zi1] that the standard action of SL(n, Z) on the n-torus is minimal in the following sense:

Conjecture 1 (Zimmer). Any smooth, volume-preserving action of any finite-index subgroup Γ < SL(n, Z) on a closed r-manifold factors through a finite group action if n > r.

While Conjecture 1 has been proved for actions which also preserve an extra geometric structure such as a pseudo-Riemannian metric (see, e.g. [Zi1]), almost nothing is known in the general case. For r = 2 and n > 4, the conjecture was proved for real-analytic actions in [FS1]. Quite recently, Polterovich [Po] has brought ideas from symplectic topology to the problem, using these to give a proof of Conjecture 1 for orientable surfaces of genus > 1. In [FS2] we will point out how his methods actually prove Conjecture 1 for the torus as well. For r = 3, Conjecture 1 is known only in some special cases where Γ contains some torsion and the action is real-analytic (see [FS1]).

In this note we prove the following result.

∗Both authors are supported in part by the NSF.

Theorem 2 (Actions on 4-manifolds with χ(M) ≠ 0). Let Γ be a lattice in a simple Lie group G such that Q-rank(Γ) ≥ 7. Then any realanalytic, volume-preserving action of Γ on a closed 4-manifold of nonzero Euler characteristic factors through a finite group action.

In particular, Theorem 2 applies to any finite-index subgroup of SL(n, Z), n > 7.

The main ingredient in the proof of Theorem 2 is Theorem 7.1 of [FS1] on real-analytic actions which preserve a volume form. This theorem, which is the most difficult result in [FS1], gives a codimension-two invariant submanifold for centralizers of elements with fixed-points. One can then apply results of [FS1] and [Re], which show that real-analytic (not necessarily area-preserving) actions of certain lattices on 2-dimensional manifolds must factor through finite groups.

For the case of symplectic actions, some further progress on Conjecture 1 can be found in [Po] and [FS2].

2 Proof of Theorem 2

Before giving the proof of Theorem 2, we will need two algebraic properties of lattices with large Q-rank.

Proposition 3 (Some algebraic properties of lattices). Let Γ be a lattice in a simple algebraic group over Q. Then the following hold:

1. If d = Q-rank(Γ) ≥ 7 then Γ contains commuting subgroups A and B which are isomorphic to irreducible lattices with Q-ranks 2 and d − 3 respectively.

2. If Q-rank(Γ) ≥ 4 then Γ contains a torsion-free nilpotent subgroup which is not metabelian.

Proof. The proof of the first statement is similar to that of Proposition 2.1 of [FS1]. By Margulis’s Arithmeticity Theorem (see, e.g., [Zi2], Theorem 6.1.2), Γ is commensurate with the group of Z-points of a simple algebraic group G defined over Q. Hence without loss of generality we can assume that Γ itself is the group of Z-points in such a group G.

Since G is simple, the root system Φ of G is irreducible, and the Dynkin diagram determined Φ therefore appears in the list given in Section 11.4 of [Hu]. By going through this list, one sees that in every case where d ≥ 7, one may “erase a vertex v” of the diagram to obtain a a graph with 2 components: one with two vertices and another which is a Dynkin diagram with at least d − 3 vertices. Let G_{1} and G_{2} be the root subgroups corresponding to these two components of the Dynkin diagram. Then the group of Q-points of G_{1} has Q-rank at least 2, the group of Q-points of G_{2} has Q-rank at least d−3, and G_{1} commutes with G_{2}.

Now Γ_{i} = Γ ∩ G_{i} is an arithmetic lattice in G_{i} for i = 1, 2, since by a theorem of Borel-Harish-Chandra (see, e.g. [Zi2]) the Z-points of an algebraic group defined over Q form a lattice in the group of R-points. Then A = Γ_{1} and B = Γ_{2} have the required properties.

To prove the second statement, note that since Q-rank(Γ) ≥ 4, we can find a connected, nilpotent Lie subgroup N which is defined over Q and has derived length 3, i.e. is not metabelian. As Γ∩N is the group of Z-points of the Q-group N, it is a lattice in N, and in particular is Zariski-dense in N. Hence Γ ∩ N is nilpotent and has no metabelian subgroup of finite index. As Γ ∩ N must have a tosrion-free subgroup of finite index, the assertion follows. ⋄

We now turn to the proof of Theorem 2. Let M be a closed 4-manifold with nonzero Euler characteristic. Let Γ be an irreducible lattice in a simple Lie group G, and assume d = Q-rank(Γ) ≥ 7. By part (1) of Proposition 3, Γ contains commuting subgroups A and B which are isomorphic to irreducible lattices with Q-ranks 2 and d − 3 ≥ 4 respectively.

Let γ_{0} be any infinite order element of A. By an old theorem of Fuller [Fu], any homeomorphism of a closed manifold of nonzero Euler characteristic has a periodic point; the proof is an application of the Lefschetz fixed-point theorem and basic number theory. Hence some positive power γ of γ_{0} has a fixed point.

We will also need the following two facts. First, since Q-rank(B) ≥ d − 3 ≥ 4, it follows from Margulis’s Superrigidity Theorem that any representation of B into GL(4, R) has finite image. Second, since Γ is a lattice in a simple Lie group G with R-rank(G) ≥ 2, the Margulis Finiteness Theorem (see, e.g., Theorem 8.1 of [Zi2]) gives that Γ is almost simple in the sense that any normal subgroup of Γ must be finite or of finite index.

We are now in a position to apply Theorem 7.1 of [FS1]. For the reader’s convenience we recall the statement here. We say that a group action ρ : Γ → Diff(M) is infinite if ρ has infinite image.

Theorem 7.1 of [FS1]: Let Γ be an almost simple group. Suppose we are given an infinite, volume-preserving, real-analytic action of Γ on a closed, connected n-manifold M. Suppose further that Γ contains commuting subgroups A and B with the following properties:

• There exists an element γ ∈ A, noncentral in Γ, having a fixed point in M.

• A is isomorphic to an irreducible lattice of Q-rank ≥ 2.

• B is noncentral in Γ.

• Any representation of any finite-index subgroup of B in GL(n, R) has finite image.

Then there is a nonempty, connected, real-analytic submanifold W ⊂ M of codimension at least 2 which is invariant under a finite-index subgroup B′ of B. Furthermore, the action of this subgroup on W is infinite.

Remark. The action of B′ on the surface W produced by this theorem is NOT necessarily area preserving.

We now conclude the proof of Theorem 2. Since B′ is a lattice in a simple Lie group and Q-rank(B′) ≥ 4, it follows from part (2) of Proposition 3 that B′ contains a torsion-free nilpotent subgroup H which is not metabelian. But Rebelo [Re] showed that any nilpotent group of real-analytic diffeomorphisms of a compact, oriented surface must be metabelian. It follows that the action of H on W is not effective.

Since H is torsion-free, there is an infinite-order element of H ≤ B′ which acts trivially on W. Since B′ has finite index in the almost simple group B, and hence is almost simple, some finite index subgroup C of B′ acts trivially on W; in particular C has a global fixed point in M. Since C is isomorphic to a lattice of Q-rank at least 4, by Lemma 3.2 of [FS1] we have that a finite index subgroup D of C acts trivially on M. Since Γ is almost simple, it follows that some finite index normal subgroup of Γ acts trivially on M, and we are done. ⋄

References

[FS1] B. Farb and P. Shalen, Real-analytic actions of lattices, Inventiones Math., Vol. 135 (1998), no. 2, p.273-296.

[FS2] B. Farb and P. Shalen, Lattice actions on symplectic 4-manifolds, in preparation.

[Fu] F.B. Fuller, The existence of periodic points, Annals of Math., Vol. 57, No. 2 (1953), p.229-230.

[Hu] J. Humphreys, Introduction to Lie Algebras and Representation Theory, GTM 9, third printing, Springer-Verlag, 1972.

[Po] L. Polterovich, Growth of maps, distortion in groups and symplectic geometry, December 2000 preprint.

[Re] J. Rebelo, On nilpotent groups of real analytic diffeomorphisms of the torus, C.R. Acad. Sci. Paris, t. 331, No. 1 (2000), p.317-322.

[Zi1] R. Zimmer, Actions of semisimple groups and discrete subgroups, Proc. I.C.M., Berkeley 1986, p.1247-1258.

[Zi2] R. Zimmer, Ergodic Theory and Semisimple Groups, Monographs in Math., Vol. 81, Birkhäuser, 1984.

Benson Farb: Dept. of Mathematics, University of Chicago 5734 University Ave. Chicago, Il 60637 E-mail: farb@math.uchicago.edu

Peter Shalen: Dept. of Mathematics, University of Illinois at Chicago Chicago, IL 60680 E-mail: shalen@math.uic.edu