Ergodic theory, groups, and geometry Informal lecture notes — not for publication

NSF-CBMS Regional Conference in the Mathematical Sciences

Ergodic Theory, Groups, and Geometry Robert J. Zimmer, principal lecturer

University of Minnesota June 22–26, 1998

These are informal lecture notes. In particular,

• the subject matter is incomplete, and

• the statements of results may be imprecise.

These notes are for personal use only. They are not intended to be accurate to the level of providing a definitive reference for citation.

Errors and omissions are solely the responsibility of Dave Witte, the editor/typist.

Revised September 29, 1998 Outline.

1. Introduction examples, problems

2. Topological theory circles, surfaces, fixed points

3. Geometric structures algebraic hulls, Lorentz manifolds

4. Fundamental groups I engaging conditions, first results

5. Gromov representation rigid geometry and π_{1}

6. Superrigidity and first applications

7. Fundamental groups II arithmetic structure

8. Locally homogeneous manifolds

9. Stationary measures projective quotients

10. Orbit structure

11. Technical appendix to Lecture 5 ideas of Gromov’s proof Ergodic theory, groups, and geometry Informal lecture notes — not for publication

1. Introduction

A group G acts on a space M.

We want to understand the relation between:

• G as a group

• the topology of M

• the geometry of M

• the “dynamics” of the action

When G is a Lie group, the structure theory tells us that there are two main cases to consider:

• solvable: start with R^{n} and use induction

• semisimple: the classification gives us a list of the groups (SL(n, R), SO(p, q), etc.), or the groups can be organized in other ways, e.g., by real rank.

A connected Lie group G often has “discrete versions:”

• lattice discrete subgroup Γ such that G/Γ is compact (or, more generally, has finite volume)

• arithmetic subgroup Let G be an algebraic subgroup of GL(n, R) and Γ = G ∩ GL(n, Z). If Γ is Zariski dense in G, then Γ is called “arithmetic.” [In other words, arithmetic groups are the Zariski-closed subgroups of GL(n, Z).]

The discrete version Γ inherits algebraic properties of G.

Assume G simple (i.e., no connected normal subgroups).

Then Γ has no infinite solvable normal subgroups (because Γ is Zariski dense in G).

If we furthermore assume R-rank(G) ≥ 2 (i.e., G has a subgroup of dimension ≥ 2 that is diagonalizable over R), then the Margulis theory links Γ and G more tightly

• Γ has no “nontrivial” normal subgroup [no infinite normal subgroup of infinite index]

• every lattice is arithmetic

Both of these conclusions can fail if we do not make an assumption like R-rank(G) ≥ 2. For example, in SL(2), the free group F_{2} is a lattice, and some other lattices have homomorphisms onto free groups.

Linear actions (finite-dimensional representations).

M = V = vector space (with dim V < ∞)

Homomorphisms G → GL(V) are classified by the theory of highest weights.

This is a classical theory.

Homomorphisms Γ → GL(V)

If R-rank(G) ≥ 2, the spirit, but not exactly the statement, of the Margulis theorem is that each representation of Γ either

• extends to a representation of G; or

• is into a compact group.

An alternate version is that (in spirit) each representation of Γ either extends to G or extends to G after some field automorphism.

Non-linear actions.

Eg. SL(n) acts on P^{n−1}

More generally, G acts on G/Q, where G/Q is a projective variety, i.e., Q = parabolic (there are only finitely many parabolics up to conjugacy).

For example, SL(n) acts on various flag varieties:

Q = ( ∗  ∗ ; ∗ ; 0  ∗ )

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

Eg. G acts on G/Γ, where Γ is a lattice More generally, if G ↪ H, and Λ is a lattice in H, then G acts on H/Λ.

Guiding question. to what extent is every action on a compact manifold (or perhaps a more general space) built out of these examples?

Construction of other examples.

• Induction

• Blowing up

Induction. Suppose H ⊂ G and H acts on a space Y. Then H acts on G × Y by h · (g, y) = (gh^{−1}, hy).

G acts on the quotient X = (G × Y)/H by a · [(g, y)] = [(ag, y)].

Ignoring the second coordinate yields a map X → G/H, and we see that X is a fiber bundle over G/H with fiber Y.

The fiber over [e] = eH is H-invariant, and the action of H on this fiber is isomorphic to the action of H on Y.

Eg. Let H be the stabilizer in SL(n) of a point in P^{n−1}, so

H = \{ h = ( \begin{matrix}λ & ∗ \\ 0 & A \end{matrix} ) | det A = λ^{−1} \}

Then h ↦ log(λ^{2}) is a homomorphism from H onto R, so every R-action yields an action of H; hence, by induction, an action of SL(n).

Upshot. Every vector field on a compact manifold M yields an SL(n)-action on a compact manifold M′.

Thus, all the complications that arise for Ractions also arise for SL(n)-actions.

It is hopeless to classify all R-actions on compact manifolds, so this discourages belief in a complete classification of actions of simple groups.

Extending these examples.

Let Γ be a lattice in SL(2) such that Γ has a homomorphism onto a nonabelian free group F.

By the same procedure as above, every action of F on a compact manifold yields an action of SL(2) on a compact manifold.

Then, in SL(n), let Q = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ), where the

box is 2×2, so Q has a homomorphism onto SL(2).

Thus, F -action → SL(2)-action → Q-action → SL(n)-action.

So the free-group problem arises for every SL(n), not just SL(2). This discourages the belief in a classification theorem even more.

Blowing up (Katok-Lewis, Benveniste).

Let Γ be a finite-index subgroup of SL(n, Z), so Γ acts on T^{n} by automorphisms.

Blow up at two fixed points x and y:

• replace x with {rays in T_{x}(T^{n})}; and

• replace y with {rays in T_{y}(T^{n})}.

Thus, x and y have each been replaced with a sphere.

Now glue the two spheres together.

Upshot. The union of two fixed points can be replaced by a projective space with the usual action of SL(n, Z) on P^{n−1}.

• this results in a different manifold (new fundamental group)

• we embedded a projective space into a volume-preserving action

• the change is on a lower-dimensional submanifold

The above is the Katok-Lewis construction. Benveniste generalized along the following lines.

Embed G in a larger group H, let M = H/Γ.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

Suppose G is contained in a subgroup L of H, such that L has a closed orbit on H/Γ.

For some pairs of closed L-orbits, we can blow up transversally (take the set of rays in a subspace transverse to the L-orbit), and then glue to make a new action.

Varying the gluing results in actions that have nontrivial perturbations.

These examples are discouraging, but interesting questions remain.

What’s left?

• Are there actions of Γ on low-dimensional manifolds?

Low-dimensional can either mean low in an absolute sense, as in dimension ≤ 3, or it can mean low relative to Γ, which means less than

(lowest dimension of representation of G) − 1

For example, if Γ = SL(n, Z), then “low-dimensional” means either ≤ 3 or < n − 1

• When are actions locally rigid?

Here, we are interested in (possibly) non-linear perturbations, even if the original action is linear.

Not all actions are locally rigid. For example, if the action is induced from a vector field, then it may be possible to perturb the vector field.

• Suppose G preserves a geometric structure on M, defined by a structure group H. Then what is the relation between G and H?

Sometimes, assuming that G preserves a suitable geometric structure eliminates the examples constructed above.

• Does every action have either:

• an invariant geometric structure of rigid type (at least, on an open set); or

• an equivariant quotient G-manifold with such a structure?

• What can be said about the fundamental group π_{1}(M), or other aspects of the topology of M?

• What is the role of the assumption of invariant volume?

Approaches to these questions must bear in mind the two constructions that may provide counterexamples to naive conjectures.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

2. Topological theory

Defn. An action of a group Γ is finite if the kernel of the action is a finite-index subgroup of Γ.

Prop. Suppose every n-dimensional linear representation of every finite-index subgroup of Γ is finite. If a smooth action of Γ on a connected nmanifold M has a finite orbit, then the action is finite.

Proof. Because Γ has a finite orbit, we may choose a finite-index subgroup Γ_{1} of Γ that has a fixed point p. Then Γ_{1} acts on the tangent space T_{p}M.

This is an n-dimensional representation of Γ_{1}, so we may choose a finite-index subgroup Γ_{2} of Γ_{1} that acts trivially on T_{p}M.

Because, by assumption, Γ_{2} has no 1-dimensional representations, we know that Γ_{2} has no nontrivial homomorphisms into R^{n}.

Then a theorem of Thurston implies that Γ_{2} fixes every point in some neighborhood of p. Thus, the fixed point set of Γ_{2} in the frame bundle of M is open. It is also closed, so we see that Γ_{2} is in the kernel of the action.

Cor. Suppose Γ is a lattice in a simple Lie group G with R-rank(G) ≥ 2. If a smooth Γaction on a circle or a surface has a finite orbit, then the action is finite.

The conclusion of the corollary can sometimes be proved without assuming that there is a finite orbit. Let us start by stating the results for actions on a circle.

Thm (Witte). Suppose Γ is an arithmetic lattice with Q-rank(Γ) ≥ 2. Then every C^{0} Γ-action on a circle is finite.

Witte’s theorem applies to any finite-index subgroup of SL(n, Z) with n ≥ 3, but, because of the assumption on Q-rank, it does not apply to any cocompact lattice in any simple Lie group. His proof is algebraic, and uses the linear order of the real line.

It should be possible to replace the Q-rank assumption with an R-rank assumption in the theorem.

Conj. Let Γ be a lattice in a simple Lie group G with R-rank(G) ≥ 2. Then every C^{0} Γ-action on a circle is finite.

Thm (Ghys, Burger-Monod). The conjecture is true for C^{2} actions.

Ghys’s proof uses the action of the homeomorphism group of S^{1} on the set of measures on the circle.

Thm (Farb-Shalen). The conjecture is true for C^{ω} actions of a special class of lattices.

The Farb-Shalen proof will be given in some detail, because this is all that we need to obtain a result for actions on surfaces.

Farb-Shalen method. The corollary at the start of this lecture assumed the existence of a finite Γorbit. In certain cases, it suffices to know that a single element γ of Γ has a finite orbit. A finite orbit of a single element can be found by topological methods.

Defn (Farb-Shalen). Let Γ be a lattice in a simple Lie group G with R-rank(G) ≥ 2.

We call Γ big if there exist

• an element γ of Γ of infinite order; and

• a subgroup Λ of the centralizer Z_{Γ}(γ), such that Λ is isomorphic to a lattice in some simple Lie group of real rank at least two.

Rem. 1) Every finite-index subgroup of SL(n, Z) is big if n ≥ 5.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

2) There exist big cocompact lattices in some simple Lie groups.

3) There are non-big lattices in simple Lie groups of arbitrarily large real rank.

Defn. We call Γ very big if there exist

• an element γ of Γ of infinite order; and

• a subgroup Λ of the centralizer Z_{Γ}(γ), such that Λ is isomorphic to a big lattice in some simple Lie group of real rank at least two.

Rem. 1) Every finite-index subgroup of SL(n, Z) is very big if n ≥ 7.

2) There exist very big cocompact lattices in some simple Lie groups.

Thm (Farb-Shalen). Assume Γ is very big. Then a C^{ω} action of Γ on a surface S is finite if either

• χ(S) ≠ 0 (Euler characteristic); or

• the action is volume-preserving.

First, let us prove Farb-Shalen’s theorem on the circle.

Thm (Farb-Shalen). Assume Γ is big. Then every C^{ω} action of Γ on the circle is finite.

Proof. Because Γ is big, we may choose γ and Λ as described in the definition. The classic Denjoy’s Theorem implies that there are two cases to consider.

Thm (Denjoy). If f is a C^{∞} diffeomorphism of S^{1}, then either

• ∃k > 0, f^{k} has a fixed point; or

• f is C^{0}-conjugate to an irrational rotation.

Case 1. Assume γ^{k} has a fixed point. Because the action is C^{ω}, the fixed points are isolated, so the fixed-point set of γ^{k} is finite. But, because Λ centralizes γ^{k}, this fixed-point set is Λ-invariant. Thus, Λ has a finite orbit. Therefore, by the corollary at the start of the lecture, the action of Λ is finite, so the kernel of the action contains a finiteindex subgroup of Λ. Because all infinite normal subgroups of Γ have finite index in Γ (Margulis’ Theorem), this implies that the Γ-action is finite.

Case 2. Assume γ is C^{0}-conjugate to an irrational rotation. Replacing γ by a conjugate, we may assume γ is an irrational rotation. The powers of γ are dense in the group of all rotations, so Λ centralizes all the rotations of S^{1}. Thus, Λ acts by rotations, so the commutator subgroup [Λ, Λ] acts trivially. Because [Λ, Λ] is infinite, we again conclude that the Γ-action is finite.

Note that the real-analyticity was used only in Case 1, to show that the fixed-point set of γ^{k} is finite. Without this assumption, the fixed-point set could be any closed subset of S^{1}.

The Farb-Shalen proof for surfaces is based on a similar idea. Given γ ∈ Γ, consider the fixed-point set M^{γ}. This is an analytic set, i.e., it is closed and defined locally by zeros of analytic functions. Analytic sets need not be manifolds (i.e., there may be singularities), but we have the following basic lemma.

Lem. If V ⊂ M is a nonempty analytic set, then ∃W ⊂ V, such that

• W is a compact analytic submanifold of V; and

• W is canonical: if f: M → M is any C^{ω} diffeomorphism, such that f(V) ⊂ V, then f(W) ⊂ W.

Farb-Shalen’s proof for surfaces. Choose γ and Λ as in the definition of very big.

Assume for the moment that γ^{k} has a fixed point, for some k > 0. Then, from the lemma, we see that Λ acts on a circle or a finite set. Because actions of big lattices on a circle (or on a finite set) are finite, we conclude that the Γ-action is finite.

Thus, we need only show that γ^{k} has a fixed point. If χ(S) ≠ 0, then this follows easily from the Lefschetz Fixed-Point Theorem. If χ(S) = 0, then it is not so easy to get a fixed point, but, by using the entire group Γ, it is possible to apply a fixed-point criterion of J. Franks.

Rem. Using the result of Ghys or Burger-Monod, we deduce the Farb-Shalen theorem for surfaces for big lattices, not only very big lattices.

3. Geometric structures

Assumption. G is a noncompact simple Lie group and M is an n-manifold.

Recall that the frame bundle P M is a principal GL(n, R)-bundle over M.

Defn. Suppose P → M is a principal H-bundle, and L is a subgroup of H. A reduction of P to L is any of the following three equivalent objects:

1) a principal L-subbundle Q of P

2) a section M → P/L

3) an H-equivariant map ϕ: P → H/L.

(iii) ⇒ (ii): for each m ∈ M, there is a unique y in the fiber of P/L over m, such that ϕ(y) = [e].

The reduction may or may not be required to have some regularity: C^{∞}, C^{0}, measurable, etc.

Defn. Let H ⊂ GL(n, R). An H-structure on M is a reduction of P M to H.

Eg. If H = O(p, q), then an H-structure on M is equivalent to a pseudo-Riemannian metric on M of signature (p, q).

(←) Given a pseudo-Riemannian metric ⟨|⟩ of signature (p, q). For each m ∈ M, let Q_{m} be the set of all frames of T_{m}M, such that the Cartan matrix

of ⟨|⟩_{m} is ( \begin{matrix}1 \\ & . & . & . \\ & & & & 1 \\ & & & & & −1 \\ & & & & & & . & . & . \\ & & & & & & & & & −1 \end{matrix} ).

Then Q is an O(p, q)-subbundle of P M.

Any action of G on M by diffeomorphisms lifts to an action on the tangent bundle T M. It also lifts to an action on the frame bundle P M. (The derivative of a diffeomorphism takes a frame on M to another frame on M.) Thus, we can speak of G-invariant H-structures on M.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

Question. If ω is an H-structure on M, then what is Aut(M, ω) (i.e., the set of diffeomorphisms preserving ω)?

Thm. Let P → M be a principal H-bundle with a G-action, and assume μ is an ergodic G-invariant measure on M.

Then there is an algebraic subgroup L of H such that

1) there is a G-invariant reduction to L;

2) there is no G-invariant reduction to any proper algebraic subgroup of L; and

3) whenever there is a G-invariant reduction to an algebraic subgroup J of H, then J contains a conjugate of L.

Rem. It is clear that the subgroup L is unique up to conjugacy.

For the proof, we recall a few basic facts.

• A topological space Y is called tame if it is second countable and T_{0}. (“T_{0}” means ∀x, y ∈ Y, ∃ open subset U of Y, such that U contains either x or y, but not both.)

• Each atomic measure on a tame space Y is supported on a single point. (“Atomic” means that Y cannot be decomposed into two sets of positive measure.)

• If f: M → Y is a G-invariant measurable function, G is ergodic on M, and Y is tame, then f is essentially constant.

• An H-action on Y is called tame if the orbit space Y/H is a tame topological space.

Eg. 1) A proper action on a Hausdorff space is tame.

2) An algebraic action on a variety is tame.

Proof of the theorem. From the descending chain condition on Zariski-closed subgroups of H, it is clear that there is a minimal algebraic subgroup L of H, such that there is a G-invariant reduction to L. This subgroup L obviously satisfies (1) and (2).

We now prove (3). Suppose ϕ_{1}: P → H/L and ϕ_{2}: P → H/J are H-equivariant and G-invariant. Let ϕ = (ϕ_{2}, ϕ_{2}): P → H/L × H/J

and ϕ̄: M → (H/L × H/J)/H. Because H acts algebraically on the variety H/L× H/J, we know that (H/L × H/J)/H is tame. We also know that G is ergodic on M, so we conclude that ϕ̄ is essentially constant. This means that the image of ϕ is essentially contained in a single H-orbit on H/L × H/J.

The stabilizer of a point in H/L × H/J is (conjugate to) L ∩ (hJh^{−1}), for some h ∈ H, so we can identify the H-orbit with H/(L ∩ (hJh^{−1})). Thus, after discarding a set of measure 0 from M, we may think of ϕ as a G-invariant H-equivariant map P → H/(L ∩ (hJh^{−1})). Thus, ϕ is a reduction of P to L ∩ (hJh^{−1}). Then, from the minimality of L, we conclude that L ∩ (hJh^{−1}) = L, so L is contained in a conjugate of J. Therefore, a conjugate of L is contained in J.

Defn. The subgroup L of the theorem (or its conjugacy class) is the algebraic hull of the G-action.

Rem. The above is the measurable algebraic hull. For actions with a dense orbit, the same argument shows there is a C^{r}-hull with the same properties, but the reductions are defined only on open, dense, G-invariant sets.

Eg. (Computation of an algebraic hull.) Suppose G acts on M, and π is a homomorphism from G to H. Define an action of G on the trivial principal bundle P = M × H by g · (m, h) = (gm, π(g)h). We will show that the algebraic hull of this action is the Zariski closure C of π(G) in H.

For any subgroup L of H, a section of the bundle P/L is a map M → H/L. Thus, the constant map Ergodic theory, groups, and geometry Informal lecture notes — not for publication

m ↦ [e] is a G-invariant section of H/C, so the algebraic hull is contained in C.

Now, let L ⊂ C be the algebraic hull of the action. By definition, there is a G-invariant reduction to L, so there is a G-invariant map f: M → C/L. Then f_{∗}μ is a G-invariant ergodic measure on C/L, so we conclude from the Borel Density Theorem that C = L.

Thm (Borel Density Theorem). Let G be a real algebraic group with no proper, normal, cocompact, algebraic subgroups.

If G acts algebraically on a variety V, then every G-invariant probability measure on V is supported on the fixed-point set.

Thm. If G preserves an H-structure and a finite measure on M, then there is (locally) an embedding σ: G → H.

Because H ⊂ GL(n, R) by definition, where n = dim M, σ defines a representation of G. This representation contains the adjoint representation of G.

Lem. If G acts on M with finite invariant measure and essentially no fixed points, then, for almost every m ∈ M, the stabilizer G_{m} of m is discrete.

Proof. For each m ∈ M, let S(m) be the Lie algebra of G_{m}. Then S(m) is a linear subspace of the Lie algebra G, so S is a map from G to a Grassmannian variety V. Because stabilizers are conjugate, it is easy to see that S: M → V is G-equivariant, where the action of G on V is via the adjoint representation. Then S_{∗}μ is an Ad G-invariant probability measure on V, so we see that the image of S is (essentially) contained in the fixed-point set of the adjoint representation. Because G is simple, the only Ad G-invariant subspaces of G are 0 and G. Because G has no fixed points, the Lie algebra of a stabilizer cannot be G, so we conclude that the Lie algebra of almost every stabilizer is 0.

Proof of the theorem. From the lemma, we know that the stabilizer of almost every point of M is discrete, so, for each point m in M, the differential of the map G → Gm yields an embedding of G in T_{m}M. Thus, T M contains, as a subbundle, the vector bundle M × G with G-action defined by g · (m, v) = (gm, (Ad g)v). From the computation of algebraic hulls done previously, we know that the algebraic hull of the action on the principal bundle associated to this subbundle is the Zariski closure of Ad G. The algebraic hull on P M, the principal bundle associated to T M, must have the algebraic hull of this subbundle as a subquotient. Since G is simple, the algebraic hull contains Ad G.

The adjoint representation of G can almost never be embedded in a representation with an invariant Lorentz metric, so we have the following corollary.

Cor. If G preserves a Lorentz metric on a compact manifold, then G is locally isomorphic to SL(2, R).

Thm (Kowalsky). If G has finite center and preserves a Lorentz structure on a (not necessarily compact) manifold M, then either:

• the G-action is not proper, and G ∼ SO(1, n) or SO(2, n); or

• the G-action is proper, and the cohomology group H^{k}(M) vanishes whenever k ≥ n − p, where p = dim G/K is the dimension of the symmetric space associated to G.

Eg. Let M = (compact) × R. Then SL(3, R) cannot act on M preserving a Lorentz structure (because H^{n−1}(M) ≠ 0).

Thm (Adams-Stuck, Zeghib). List (up to local isomorphism) of all Lie groups that can act on a compact Lorentz manifold.

Adams-Stuck and Zeghib also list (up to local isomorphism) all possibilities for the full isometry group of a compact Lorentz manifold.

4. Fundamental groups I

This lecture is an introduction to how to understand the fundamental group.

Namely, G acts on M, and we are interested in the relation between G and π_{1}(M).

The universal cover M̃ is a principal bundle over M, with fiber π_{1}.

If Λ is a subgroup of π_{1}, then M̃/Λ is another cover of M.

There are measurable sections M → M̃/Λ; there are also sections that are continuous on an open dense set.

By a reduction of M̃ to Λ, we mean such a measurable section.

Eg. M = T^{2} = R^{2}/Z^{2}.

The universal cover M̃ is R^{2}. A section σ: R^{2}/Z^{2} → R^{2} is given by σ([(x, y)]) = ({x}, {y}), where {·} denotes the fractional part.

Suppose G acts on M. (For simplicity, assume G is simply connected, so G also acts on M̃.)

The action is Λ-reducible if there is a Gequivariant reduction to Λ.

Rem. There is no “discrete hull” because the set of discrete subgroups usually does not have the descending chain condition.

Eg. Let G be a connected, simply connected group that acts on the unit square [0, 1]^{2}, fixing each point on the boundary.

• By gluing opposite sides of the square, we obtain an action of G on T^{2}.

• The lift to an action on R^{2} is a tiling by copies of the original action: for n ∈ Z^{2} and x ∈ [0, 1]^{2}, we have g · (n + x) = n + gx.

This action is {e}-reducible.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

An “engaging condition” is a property that prohibits this type of reduction.

Defn. Let G act on a manifold M.

1) The action is totally engaging if there is a reduction to Λ ⊂ π_{1} only for Λ = π_{1}. In other words, there are no G-equivariant sections M → M̃/Λ for any proper subgroup Λ of π_{1}.

2) The action is engaging if there is reduction to Λ ⊂ π_{1} only if Λ is profinitely dense in π_{1}, i.e., Λ surjects into every finite quotient of π_{1}. (This means that Λ is a “very big subgroup” of π_{1}.)

3) The action is topologically engaging if there is reduction to Λ ⊂ π_{1} only if Λ is infinite.

Usually an action is ergodic for a good reason that is insensitive to finite covers. For example, Anosov flows are ergodic, and the Anosov condition, being local, is not affected by passing to a finite cover. So the following proposition shows that engaging is a natural assumption from a dynamic point of view.

Prop. Assume G is ergodic on M. The G-action on M is engaging iff G is ergodic on every finite cover of M.

The following is an easy way to show that an action is topologically engaging.

Prop. If G (or even a single element of G) acts properly on an open dense subset of M̃, then the action is topologically engaging.

As an example of how to use the engaging conditions, we prove the following result.

Thm. Suppose G acts on M with finite invariant measure, and satisfies any of the engaging conditions. If R-rank(G) ≥ 2, then π_{1}(M) cannot be amenable (unless π_{1}(M) is finite).

Proof. For convenience, let Γ = π_{1}. Γ acts on l^{2}(Γ) (“left regular representation”)

Defn. To say that Γ is amenable means that the regular representation has “almost invariant vectors.” That is, for every finite subset F of Γ, and every ϵ > 0, there is a unit vector v in l^{2}(Γ), such that ∀γ ∈ F, we have ||γv − v|| < ϵ.

(For example Z is amenable, because the normalized characteristic function of a long interval moves very little under translations of a bounded size.)

We can view L^{2}(M̃) as the sections of a bundle over M with fiber l^{2}(Γ). Now M̃ is a bundle over M with fiber Γ. The group G acts by bundle automorphisms, so, when an element of G maps one fiber to another, the only twisting of the fiber is via multiplication by an element of Γ. Therefore, the map from fiber to fiber in L^{2}(M̃) is via translation by an element of Γ.

Hence, by choosing a constant section M → l^{2}(Γ) whose value is an almost invariant vector in l^{2}(Γ), we see without too much difficulty that L^{2}(M̃) has almost invariant vectors. (Because M has finite measure, constant functions are in L^{2}.)

On the other hand, because R-rank(G) ≥ 2, we know that Γ has Kazhdan’s property T, which means that if a unitary representation of Γ has almost invariant vectors, then the representation has invariant vectors.

Therefore, there is a G-invariant section of the bundle over M whose fiber is the regular representation l^{2}(Γ).

Now, for any group X, the regular representation yields an action of X on L^{2}(X) − {0}, and this is always a proper action. Thus, using the same tameness arguments as yesterday, we obtain a section of a bundle whose fiber is a single Γ-orbit, and we can identify this orbit with Γ/Γ_{0}, where Γ_{0} is Ergodic theory, groups, and geometry Informal lecture notes — not for publication

the stabilizer of some nonzero vector f ∈ l^{2}(Γ). This is exactly a reduction to Γ_{0}. Because the stabilizer of every nonzero vector in l^{2}(Γ) is finite, we know that Γ_{0} is finite.

Thus, we have a reduction to a finite subgroup of π_{1}. This contradicts all of the engaging hypotheses.

The amenability hypothesis was used to obtain almost invariant vectors in the regular representations. For some nonamenable groups, there are other representations that have almost invariant vectors.

Prop. Let H = SL(2, R), or SO(1, n) or SU(1, n). Then there is a unitary representation (H, π) of H, such that

1) H acts properly on H − {0}; and

2) π has almost invariant vectors.

Statement (1) follows from the vanishing of matrix coefficients for unitary representations of semisimple groups [R. Howe, C. C. Moore, T. Sherman]. The existence of representations satisfying Statement (2) is a consequence of the fact that H does not have Kazhdan’s property T.

By restricting π, we obtain a representation of any closed subgroup of H with the same properties. Thus, the proof of the theorem yields the following result:

Thm. Suppose G acts on M with finite invariant measure, and satisfies any of the engaging conditions. If R-rank(G) ≥ 2, then π_{1}(M) cannot be an infinite discrete subgroup of SO(1, n) or SU(1, n). In particular, π_{1}(M) cannot be a surface group, a free group, or the fundamental group of a real hyperbolic manifold.

Eg. Actions with engaging conditions.

Let M = H/Γ, where G embeds in H as a closed subgroup.

i) The action is topologically engaging. Because G acts properly on H, it acts properly on the universal cover of H, which is the same as the universal cover M̃ of M. Therefore, it is obvious that the action is topologically engaging.

ii) The action is engaging. Any finite cover of M is a coset space H/Γ′, for some finite-index subgroup Γ′ of Γ. Because Γ′ is a lattice in H, the Moore Ergodicity Theorem implies that G is ergodic on H/Γ′. Therefore, the action is engaging.

iii) The action is totally engaging. A cover of M is a coset space H/Λ, for some subgroup Λ of Γ. (Note that Λ can be any subgroup of Γ, perhaps of infinite index, so it might not be a lattice in H.) Suppose s: H/Γ → H/Λ is a section.

Because Γ is a lattice in H, there is an H-invariant probability measure μ on H/Γ. The Moore Ergodicity Theorem implies that μ is G-ergodic, so s_{∗}μ is an ergodic G-invariant probability measure on H/Λ.

Ratner’s Theorem implies that there is a closed, connected subgroup L of H, such that s_{∗}μ is supported on a single L-orbit. Because s_{∗}μ must project to μ, which gives measure 0 to any immersed submanifold of lower dimension, we must have L = H.

However, Ratner’s Theorem also tells us that s_{∗}μ is L-invariant, so, because L = H, we conclude that s_{∗}μ is H-invariant. Thus, H/Λ has an Hinvariant probability measure, namely s_{∗}μ, so Λ is a lattice in H.

Therefore, the Moore Ergodicity Theorem implies that G is ergodic on H/Λ, so, because there is a G-invariant section H/Γ → H/Λ, it is easy to see that H/Λ cannot be a proper cover of H/Γ. Therefore, Λ = Γ.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

Gromov provided another important class of topologically engaging examples. We will see the proof later.

Thm (Gromov). If G preserves a connection and a volume form on M, and everything is real analytic, then the action is topologically engaging.

Eg. The Katok-Lewis and Benveniste examples constructed by blowing up are not engaging. For example, there is a double-cover of the Katok-Lewis example that, after throwing out a set of measure 0, consists of two copies of the original action. (The set of measure 0 consists of the spheres along which the two copies are glued together.) Therefore, the double cover is not ergodic, so the action is not engaging.

Here is a construction of such a double cover. The Katok-Lewis example M is constructed from a certain space X (obtained by blowing up two fixed points) that contains two distinguished spheres A and B, by gluing A to B. Let X′ be a copy of X, with distinguished spheres A′ and B′. Now glue A to B′ and glue A′ to B, resulting in a space Y that is a double cover of M. (Identifying X with X′ collapses Y to M.) If we remove A and B from Y, then the gluing is also removed, so we obtain X and X′ with their distinguished spheres removed.

On the other hand, examples of Katok-Lewis-Benveniste type can be topologically engaging, because there may be covers on which the action of G is proper. In particular, if the original manifold (before the blowing up) has a cover on which G acts properly, then this yields a cover of the Katok-Lewis-Benveniste example on which G acts properly.

This illustrates the difference between engaging and topological engaging. For an action to be topologically engaging, there only needs to be some subgroup of the fundamental group that yields a proper action, and the rest of the fundamental group is irrelevant. In contrast, the condition for engaging sees all of the fundamental group (if π_{1} is residually finite).

5. Gromov representation

Suppose G acts on a manifold M. Let M̃ be the universal cover of M and let Γ = π_{1}(M).

We can try to understand Γ by understanding its representation theory. That is, by understanding Hom(Γ, H), where H is an algebraic group, for example, H = GL(n).

Given σ ∈ Hom(Γ, H), we can form the associated principal bundle P_{σ} over M with fiber H:

P_{σ} = (M̃ × H)/Γ.

G acts on P_{σ}. Let L ⊂ H be the algebraic hull of this action, and let \overline{\overline{σ(Γ)}} be the Zariski closure of σ(Γ). Because (M̃ × \overline{\overline{σ(Γ)}})/Γ is a G-invariant subbundle of P_{σ}, which means P_{σ} has a reduction to \overline{\overline{σ(Γ)}}, we know that L is contained in \overline{\overline{σ(Γ)}}.

Upshot. If we can find a representation σ of Γ, such that the algebraic hull of P_{σ} is “large,” then the Zariski closure of σ(Γ) must be large.

Thus, we have a method for showing that there is a representation σ of Γ, such that the Zariski closure of σ(Γ) is large. This can be a significant restriction on the possibilities for the fundamental group Γ. For example, a superrigid lattice in a group H does not have any homomorphism whose image has a large Zariski closure, because the Zariski closure of the image is basically H (but, maybe with a compact group added on).

We know how to calculate the algebraic hull in some cases; if we can do the calculation for these associated bundles for appropriate representations σ, then we will obtain information about the fundamental group of M.

The fundamental group has a natural action on Vect(M̃), the space of vector fields on M̃. However, this is an infinite-dimensional representation, so the preceding comments do not apply.

Suppose there is a subspace V of Vect(M̃), such that 1) V is finite-dimensional; and

2) V is Γ-invariant.

The evaluation map ev: M̃×V → T M̃ is a Γ-map.

Passing to the quotient, we obtain a vector-bundle map ẽv: (M̃ × V)/Γ → T M. If G acts trivially on V, i.e., if 3) V centralizes G

(where we think of the Lie algebra G as a subspace of Vect(M̃), via the action of G on M̃), then

ẽv is a G-equivariant map.

If we also have

4) ẽv((M̃ × V)/Γ) ⊃ T (G-orbits), then the algebraic hull of the action on (M̃×V)/Γ must contain G. (This follows from the Borel Density Theorem as in the proof that if G preserves an H-structure, then G embeds in H, and this representation contains the adjoint representation of G, coming from the G-orbits.)

Thm (Gromov’s Centralizer Theorem). Assume • G acts on M;

• M has a connection and a volume form, each of which is C^{ω} and G-invariant;

• M is compact; and

• G is simple and noncompact (but not necessarily of higher rank). Let V be the set of all vector fields X in Vect(M̃), such that

a) X centralizes G and

b) X preserves the connection. Then

1) V is Γ-invariant;

2) dim V < ∞;

3) V centralizes G; and

4) ∀x ∈ M̃, ev_{x}(V) ⊃ T Gx.

Condition (a) means that X is G-invariant. Condition (b) means that the connection is invariant Ergodic theory, groups, and geometry Informal lecture notes — not for publication

under the local one-parameter group generated by X.

Conclusion (1) is obvious because Γ centralizes G and the connection is Γ-invariant (being lifted from a connection on M).

Conclusion (2) is the “rigidity of connections.” It follows from the fact (which will be discussed in more detail later in these lectures) that any connection defines a Riemannian metric on the frame bundle, and the fact that the isometry group of a Riemannian manifold is finite dimensional.

Conclusion (3) is immediate from the definition of V.

Conclusion (4) is not at all obvious; it is not even clear that there is a single nonzero vector field that commutes with G.

The proof of the theorem is difficult, because of Conclusion (4); it will be postponed to a later lecture. The crucial point is the computation of the algebraic hull by using the Borel Density Theorem not just in the tangent bundle, but also in higher order frame bundles.

Eg. Suppose M = G/Γ, so M̃ = G. Then G is the space of left-invariant vector fields on G, so V is the space of right-invariant vector fields on G.

From the argument described before the statement of the theorem, we have the following corollary.

Cor. There is a representation (namely, V), such that \overline{\overline{σ(Γ)}} contains a subgroup locally isomorphic to G.

Cor. Suppose G = SL(4, R). Then π_{1}(M) cannot be a lattice in SL(3, R).

Proof. Let Γ be a lattice in SL(3, R), and let σ be any representation of Γ. The Margulis Superrigidity Theorem implies that the Zariski closure of the image of σ is essentially SL(3, R) × (compact) — it certainly does not contain SL(4, R).

Cor. Assume

• G acts on M;

• M has a connection and a volume form, each of which is C^{ω} and G-invariant;

• M is compact; and

• G is simple and noncompact (but not necessarily of higher rank). Then the G-action is topologically engaging.

Proof. Let P be the principal bundle associated to V, and let Q be the principal bundle associated to T (G-orbits). There is an embedding of M̃ in P, and Q is a quotient of a subbundle of P.

The bundle Q contains M × G, where G acts diagonally: a · (m, g) = (am, ag). Therefore, G is obviously proper on Q. This ultimately leads to properness on M̃.

As background to the proof of Gromov’s Theorem, we now discuss how to think of a connection in terms of H-structures. As mentioned in a previous lecture, a pseudo-Riemannian metric is more-or-less obviously equivalent to an O(p, q)-structure. A connection is an H-structure for a higher-order frame bundle.

Higher-order frames and connections.

Notation. G(R^{n}, 0; M) is the set of functions R^{n} → M defined near 0 (“germs”)

f ∼_{k} h if the derivatives up to order k agree.

An equivalence class is a k-jet.

J^{k}(R^{n}, 0; M) is the set of all k-jets

P^{k}(M) is the subset of J^{k}(R^{n}, 0; M) consisting of the k-jets of local diffeomorphisms

Thus, P^{k}(M) is a bundle over M.

• The map P^{k}(M) → M evaluates a k-jet at 0.

• The fiber over m is the set of all k-jets of local diffeomorphisms (R^{n}, 0) ↔ (M, m). Each point in the fiber is a “kth-order frame at m.”

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

• P^{k}(M) is called the kth order frame bundle.

Eg. Consider P^{1}(M). For [f] ∈ P^{1}(M), the derivative df_{0}: R^{n} → T M_{m} is an isomorphism, so we see that P^{1}(M)_{m} is the set of all linear isomorphisms from R^{n} to T M_{m}. Thus, we see that P^{1}(M) is the frame bundle P (M).

Defn. Let GL^{(k)} be the set of all k-jets of local diffeomorphisms (R^{n}, 0) ↔ (R^{n}, 0).

• GL^{(1)} = {invertible 1st derivatives} = GL(n, R).

• The map f ↦ (df_{0}, D^{2}f_{0}) yields a bijection from GL^{(2)} onto GL(n, R) × S, where S = S^{2}(R^{n}; R^{n}) is the space of symmetric R^{n}-valued 2-tensors on R^{n}. This yields a realization of GL^{(2)} as a semidirect product GL(n, R) × S, for some action of GL(n, R) on the abelian group S.

• Exercise. GL^{(k)} ≅ GL(n, R) × U, for some unipotent group U.

GL^{(k)} acts on P^{k}(M) (on the right), so we see that P^{k}(M) is a principal GL^{(k)}-bundle. Note that P^{k}(M) is a natural bundle, by which we mean that every diffeomorphism of M yields a diffeomorphism of P^{k}(M).

For H ⊂ GL^{(k)}, we can consider “higher-order Hstructures.”

Prop. A torsion-free connection on M is equivalent to a reduction of the GL^{(2)}-bundle P^{2}(M) to GL(n, real).

Proof. (⇐) Suppose we have a reduction of P^{2}(M) to GL(n, R).

This means that, for each m ∈ M, we have a 2jet [ϕ]: (R^{n}, 0) → (M, m); the 2-jet is not welldefined, but it is well-defined up to composition with an element of GL(n, R), because the reduction is to GL(n, R).

Given X, Y ∈ Vect(M) near m, we want to define the covariant derivative D_{X}Y. Let D_{X}Y = ϕ_{∗}(∇_{ϕ_{∗}^{−1}X}(ϕ_{∗}^{−1}Y)),

where ∇ is the ordinary covariant derivative on R^{n}.

The value of D_{X}Y does not change if ϕ is replaced with ϕA, for any A ∈ GL(n, R), so D is well defined.

Prop. Suppose G acts on M, and let L^{(2)} ⊂ GL^{(2)} be the algebraic hull of the action of G on P^{2}(M). If L^{(2)} is reductive, then there is a measurable G-invariant connection.

(If we use the C^{r}-algebraic hull instead of the measurable hull, then there is a connection that is C^{r} on an open dense subset of M.)

Proof. Any reductive subgroup of GL^{(2)} is contained in (a conjugate of) GL(n, R).

R. J. Zimmer proved that if R-rank(G) ≥ 2, then the measurable algebraic hull for any bundle is reductive with compact center. Therefore, the proposition implies that if R-rank(G) ≥ 2, then every action of G has a measurable invariant connection.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

6. Superrigidity and first applications

Suppose G acts on M, and the action lifts to an action on a principal H-bundle P over M. The cocycle superrigidity theorem is an explicit description of these actions in the case where H is algebraic and R-rank(G) ≥ 2.

Any section s of P yields a “trivialization” of P, that is, an isomorphism of P with the trivial Hbundle M × H. The isomorphism from M × H to P is given by (m, h) ↦ s(m) · h.

Question. Can we choose s so that the action of G is also very simple?

Given m ∈ M and g ∈ G, the two points s(gm) and g · s(m) are both in the fiber over gm, so they differ by an element of H; there is an element h(g, m) ∈ H with s(gm) = g · s(m) · h(g, m).

Defn. Let π be a homomorphism from G to H. We call the section s totally π-simple if s(gm) = g · s(m) · π(g)^{−1} for every g ∈ G and m ∈ M.

Another way of saying this is that, under the trivialization M × H ↔ P defined by s, the G-action on M × H is given by g · (m, h) = (gm, π(g)h). Thus, a totally π-simple section “trivializes” the G-action.

[The function (g, m) ↦ h(g, m) is a cocycle. The search for a totally π-simple section is exactly the search for a homomorphism cohomologous to h.]

Eg. Suppose G acts locally freely on M, and let P be the frame bundle of T (G-orbits). Then there is a smooth totally Ad-simple section of P. To see this, note that, for each x ∈ M, the map G → Gx yields an isomorphism of the Lie algebra G with T (G-orbit)_{x}, because G acts locally freely (i.e., all stabilizers are discrete). Therefore, letting E = M × G, and choosing a basis of G, or in other words, choosing an identification of G with R^{n}, we have an isomorphism from M × R^{n} to T (G-orbit). The G-action on E is given by g · (m, x) = (gm, (Ad g)x). We then clearly have a similar statement for P.

Eg. Suppose Γ is a lattice in G and σ: Γ → H is a homomorphism. Let P = P_{σ} = (G × H)/Γ, a principal H-bundle over G/Γ. It is not difficult to show that there is a homomorphism π: G → H and a totally π-simple section of P iff σ extends to a homomorphism π: G → H. (The same homomorphism π can be used in both statements.)

Thus, in this example, the question of whether the action on the principal bundle can be trivialized is precisely asking whether a homomorphism defined on Γ can be extended to be defined on all of G.

Weaker definition. A section s: M → P is πsimple if there is a compact subgroup C of H that centralizes π(G), such that, for every g ∈ G and m ∈ M, there is some c(g, m) ∈ C, such that s(g, m) = g · s(m) · π(g)^{−1} · c(g, m). We think of c(g, m) as being “compact noise,” that is, a small deviation from the ideal result.

Thm (Zimmer Cocycle Superrigidity Theorem). As usual, assume G acts on a compact space M, and the action lifts to a principal H-bundle P over M, with H an algebraic group.

Assume also that R-rank(G) ≥ 2 and that G is ergodic on M with finite invariant measure.

Then there is a homomorphism π: G → H and a measurable π-simple section.

[More precisely, H is a real algebraic group, or more generally, the k-points of a k-group, where k is a local field of characteristic 0.]

(The same theorem applies to Γ-actions, where Γ is a lattice in G.)

In the case where P = P_{σ} is the principal bundle defined above from a homomorphism σ: Γ → H, the cocycle superrigidity theorem reduces to the Margulis Superrigidity Theorem.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

[In terms of cocycles, the theorem says that every cocycle is cohomologous to a homomorphism times a “noise” cocycle with precompact image.]

[For homomorphisms of lattices, the Margulis Superrigidity Theorem also describes the homomorphisms into compact groups (in terms of field automorphisms). Unfortunately, for a general action on a principal bundle, we do not have a good description of the cocycles into compact groups (the “compact noise”). The main obstacle to generalizing Margulis’ results to these cocycles is the question of whether or not the cocycle is cohomologous to a cocycle with countable image.]

The superrigidity theorem yields only a measurable section. The lack of smoothness is a shortcoming in applications to geometric questions.

Conj. Suppose Γ acts on a manifold M, and dim M is less than a certain function of the dimension of the smallest nontrivial representation of G. Then Γ acts by a finite quotient.

Thm (Farb-Shalen). The conjecture is true in the C^{ω} category for certain lattices when dim M = 2.

Here is an example of another setting where one can prove the conclusion of the conjecture:

Thm. The conjecture is true for connectionpreserving actions.

Conj. If dim M < n, then any volume-preserving action of SL(n, Z) on M is finite.

The conjecture is true for connection-preserving actions. Note that the conjecture is sharp, because SL(n, Z) acts on T^{n}.

Proof of the theorem. Since dim M is small, the homomorphism π in the conclusion of the Cocycle Superrigidity Theorem is trivial. Thus, there is a compact subgroup C of H, such that, for every γ ∈ Γ and m ∈ M, we have s(γm) ∈ γ · s(m)C. Therefore, s(M)C is a Γ-invariant subset of P.

We claim that

(1) there is a finite Γ-invariant measure on P. This is obvious if C is trivial, because then s is a Γ-equivariant embedding of M into P, so the invariant measure on M yields an invariant measure on P. In general, s is a non-equivariant embedding of M into P, so the measure on M yields a non-invariant measure on P. However, averaging the translates of this measure by elements of C yields an invariant measure.

Simple fact. A connection on M defines a natural Riemannian metric on P.

Because the connection is Γ-invariant and the metric is natural, we see that

(2) Γ preserves a Riemannian metric on P.

Statements (1) and (2) are also true for the closure \overline{Γ} of Γ, which implies that \overline{Γ} is a compact Lie group. Then the Margulis Superrigidity Theorem implies that dim \overline{Γ} is “large.” Because there is an explicit bound on the dimension of the isometry group of a Riemannian manifold in terms of the dimension of the manifold, we conclude that dim M is “large.”

The functions we have called “large” are explicit and quite reasonable. For example, consider the result described above for SL(n, Z).

Thm (Zimmer, Feres). Suppose SL(n, Z) acts C^{∞} on a compact manifold M of dimension n, preserving a connection and a volume form. Then M = T^{n} with the standard connection, and the action is standard (or the contragredient).

Sketch of proof. From the superrigidity theorem, we have a homomorphism π: SL(n, R) → SL(n, R). If π is trivial, then we use the same proof as before. Thus, we may now assume π is the identity map (or the contragredient). Therefore, the action on P is measurably isomorphic to the standard action. Use the connection to get a smooth isomorphism from the measurable isomorphism.

The preceding theorem is one example of using a geometric structure to turn a measurable structure into something better. In this case, the geometric structure is a connection. Another useful structure is hyperbolicity, which has been studied in this context by many persons (e.g., Hurder, Katok, Lewis, Zimmer, Spatzier, Goetze, Feres, Yue, Qian, and Margulis).

Another approach to getting a conclusion that is better than measurable would be to modify the proof of the superrigidity theorem itself to get a better conclusion in some settings. The proof of the Margulis Superrigidity Theorem starts by using the Multiplicative Ergodic Theorem (or amenability, or, in Furstenberg’s approach, random products) to get a measurable map. Then machinery makes the map better. All ways (such as the Multiplicative Ergodic Theorem, amenability, and random products) to create a map from nothing seem to give measurable maps, because it is easier to construct a measurable map. The Katok-Lewis and Benveniste counterexamples show that one cannot expect to get a smooth section under very general conditions. However, in some reasonable generality, one can hope to get a smooth section on an open set.

There are also situations in which one can use Sobolev space techniques to improve regularity of the measurable structure.

7. Fundamental groups II (Arithmetic theory)

Recall. We study π_{1} via Hom(π_{1}, H), where H is an algebraic group. Given σ ∈ Hom(π_{1}, H), we apply the cocycle superrigidity theorem to the G-action on the associated principal H-bundle P_{σ} over M. Then, by using engaging properties, we deduce results about σ. Often, we obtain enough information about π_{1} to determine many properties. The goal is to describe π_{1} in terms of arithmetic groups.

Defn. A Q-group H is a subgroup of GL(n, C) defined by zeros of polynomials with coefficients in Q. An arithmetic group is a group commensurable to H ∩ GL(n, Z), for some Q-group H. Equivalently, one may say that an arithmetic group is a group that is commensurable to a Zariski-closed subgroup of GL(n, Z) (where GL(n, Z) inherits a Zariski topology because it is a subset of the variety GL(n, C).

Defn. Let S be a finite set of prime numbers, and let Z_{S} = Z[1/p_{1}, …, 1/p_{r}], where S = {p_{1}, p_{2}, …, p_{r}}.

An S-arithmetic group is of the form H ∩ GL(n, Z_{S}) for some Q-group H.

An s-arithmetic group is a subgroup Γ of H ∩GL(n, Z_{S}), such that Γ contains H ∩GL(n, Z), for some Q-group H.

When H is semisimple, Venkataramana showed that any s-arithmetic group is S′-arithmetic, for some S′ ⊂ S. However, we do not want to assume H is semisimple, because, even though G is semisimple, it can be embedded in a nonsemisimple group H, and G acts on H/Γ.

If H = [H, H], then any s-arithmetic subgroup Γ of H is a lattice in some locally compact group L.

Namely, L is the closure of ΓH_{R} in H_{R}×∏_{i=1}^{r} H_{Q_{pr}}.

Thm (Margulis Arithmeticity Theorem). Any irreducible lattice in a semisimple group of real rank Ergodic theory, groups, and geometry Informal lecture notes — not for publication

at least two is arithmetic.

In this lecture, M is any second countable, metrizable topological space that satisfies the usual covering space theory; in particular, M does not need to be a manifold.

Thm (Zimmer, Lubotzky-Zimmer). Suppose G acts on M with finite invariant measure μ, and assume R-rank(G) ≥ 2. Let σ: π_{1}(M) → GL(n, C), and assume Γ = σ(π_{1}(M)) is infinite.

• If the action is totally engaging, then Γ is arithmetic (in some Q-group H that contains G).

• If the action is engaging, then Γ is sarithmetic (in some Q-group H that contains G).

• If the action is topologically engaging, and σ is injective with discrete image, then Γ contains an arithmetic subgroup of some Q-group H that contains G.

Rem. The group H is closely related to the Zariski closure of σ(Γ), but it is not exactly the same group. They have the same Levi subgroup, and the same representation of the Levi subgroup on the Lie algebra, but their unipotent radicals may not be isomorphic.

Rem. To understand Γ, we need to take many representations: not only real representations, but also p-adic representations and representations over fields of positive characteristic. Even when M is a manifold, we need to analyze representations of positive characteristic.

Open question. If M is a manifold, and we assume the action is engaging, can π_{1} really be sarithmetic without being arithmetic?

If M is not a manifold, it is easy to construct actions where the “fundamental group” is Sarithmetic, but not arithmetic. For example, G acts on (G × Building)/Γ, where Γ is an Sarithmetic group. This action is engaging, but not totally engaging.

One main idea in the proof. For simplicity, let us assume σ is the identity map, so π_{1}(M) = Γ is a discrete subgroup of a real algebraic group H in GL(n, R). Applying the cocycle superrigidity theorem to the associated principal H-bundle P_{H} = (M̃ × H)/Γ yields a section s: M → P_{H}, a homomorphism π: G → H, and a compact subgroup C of H, such that

s(gm) = g · s(m) · π(g)^{−1} · c(g, m). Also, because P_{H} = (M̃ × H)/Γ, there is an Hequivariant projection q: P_{H} → Γ\H.

Let φ = q ∘ s: M → Γ\H, so

φ(gm) = φ(m) · π(g)^{−1} · c(g, m). By replacing the right H-action on Γ\H with the left H-action on H/Γ, we obtain a map ψ: M → H/Γ with

ψ(gm) = c(g, m)^{−1} · π(g) · ψ(m). Mod out C to get θ: M → C\H/Γ. Then θ(gm) = π(g) · θ(m), so θ is a G-map.

Therefore, θ_{∗}(μ) is a G-invariant measure on C\H/Γ. (However, this measure may be singular with respect to the Haar measure on C\H/Γ; the image of θ may be a null set.) Ratner’s Theorem tells us there is a subgroup L of H containing G, such that Γ ∩ L is a lattice in L, and (a translate of) θ_{∗}(μ) is the Haar measure on C′\L/(Γ ∩ L).

Upshot. Γ contains a lattice (namely, Γ ∩ L) in a group (namely, L) that contains G.

This is precisely the desired conclusion when the action is assumed to be topologically engaging. For other engaging hypotheses, stronger conclusions are required, so further arguments are needed to get Γ closer to being an arithmetic group, instead of just containing an arithmetic group.

We assumed σ(Γ) = Γ is discrete. In general, some standard algebraic methods are used to convert σ(Γ) to a discrete group.

Filling a gap in the proof. The above argument is incomplete, because we did not show that π is nontrivial. (If π(G) = e, then we only showed that Γ ⊃ e, which is obviously not an interesting conclusion.) This is where we use the engaging hypothesis.

Suppose π is trivial. Recall that this implies we have a section s: M → P_{H} with s(gm) = g · s(m) · c(g, m), so we have a G-map M → P_{H}/C. This means we have a G-invariant reduction of P_{H} to C, so there is a G-invariant H-equivariant map P_{H} → H/C. From the definition of P_{H}, we also have a G-invariant H-equivariant projection P_{H} → H/Γ. Thus, we have a G-invariant H-equivariant map τ: P_{H} → H/C × H/Γ.

Because C is compact, the action of H on H/C is proper, so the action of H on H/C×H/Γ is proper, hence tame. Thus, because G is ergodic on M, we conclude that the image of τ is contained in a single H-orbit, which we may identify with H/(Γ∩ hCh^{−1}) for some h ∈ H. Because Γ is discrete and hCh^{−1} is compact, we know Γ ∩ hCh^{−1} is finite. Thus, τ is a G-invariant reduction of P_{H} to a finite subgroup of Γ. Since P_{H} comes from M̃, then it is not hard to show that M̃ also has a Ginvariant reduction to a finite subgroup of Γ. This contradicts any engaging hypothesis.

8. Locally homogeneous spaces

This lecture is not intended to be a survey of locally homogeneous spaces, but just an indication of how two techniques of ergodic theory can be applied to handle a number of particular directions in the subject.

Defn. Suppose M andX are spaces, and G is a pseudogroup of local homeomorphisms of X.

We say that M is locally modeled on (X, G) if there is an atlas of local homeomorphisms from open sets in M to open sets in X, such that the transition functions in X are elements of G.

Eg. Let X = R^{n}, and let G be the set of all local diffeomorphisms that preserve the R^{k}-direction.

Then a space modeled on (X, G) is a foliated manifold.

Basic case. Let X = G/H, where G is a Lie group and H is a closed subgroup, and let G consist of the restrictions of elements of G to open sets in X.

Defn. If M is locally modeled on (G/H, G), we say M is locally homogeneous and M is a form of G/H.

Eg. If G = SO(1, n) and X = H^{n}, then the forms of X are the hyperbolic manifolds.

The definition of a hyperbolic manifold is not in terms of curvature — it is a theorem that if M is a Riemannian manifold of constant curvature −1, then M is of this form.

Eg. Let G = Aff(R^{n}) ≅ GL(n, R) × R^{n}, and let X = Aff(R^{n})/ GL(n, R) ∼ R^{n}. Then the forms of X are the affinely flat manifolds.

Problem. Given X = G/H, understand the forms of G/H.

• For example, find all compact forms.

• When does there exist a compact form?

Special case of forms. If there is a discrete subgroup Γ of G, such that Ergodic theory, groups, and geometry Informal lecture notes — not for publication

• Γ acts properly discontinuously on G/H; and

• Γ\G/H is compact, then M = Γ\G/H is a compact form of G/H.

This is called a “complete form” or a “compact lattice form.”

Even when G/H is a symmetric space, it was once a question whether or not there always exists a compact form, but this case was settled by Borel:

Thm (Borel). If G is semisimple and H is compact, then there is a compact lattice form of G/H.

Rem. When H is compact, the conditions on the subgroup Γ for Γ\G/H to be a compact lattice form are equivalent to the conditions for Γ to be a cocompact lattice.

On the other hand, when H is not compact, a lattice is never properly discontinuous on G/H (for example, this follows from the Moore Ergodicity Theorem), so we need to look for other kinds of discrete subgroups.

General belief. Suppose G is simple and H is a closed, noncompact subgroup. Then it is unusual for G/H to have a compact lattice form, or even to have a compact form. (There do exist examples of compact forms, but the feeling is that there should not be many of them.)

If H is compact, then all compact forms of G/H are compact lattice forms, but, in general, the compact forms can be a significantly larger class than the lattice forms, as is illustrated by the following example.

Eg. Let G = GL(n, R). Because G is reductive (essentially, G = (semisimple) × R), it is not difficult to see that there is a lattice Λ in G. Let M = G/Λ.

Thinking of G as an open subset of R^{n×n}, it is easy to see that multiplication by elements of Λ preserves the affine structure on R^{n×n}, so M is affinely flat. This is not a double-coset space of Aff(R^{n}). It is believed that the fundamental group of a compact lattice form must be amenable, but the fundamental group Λ in our example is far from being amenable.

Conj (L. Auslander). If M is a compact lattice form for

Aff(R^{n})/ GL(n, R) ≅ (R^{n}, as an affine space), then π_{1}(M) is amenable.

The plausibility of the conjecture, contrasted with the example, shows at least the potential for lattice forms and general forms to be very different. Therefore, when it has been shown that a homogeneous space G/H does not have any compact lattice forms, the existence of a general compact form remains an interesting question.

Rem. Milnor conjectured that the compactness assumption on M could be eliminated, but Margulis found a counterexample to this stronger conjecture.

Test cases for the General Belief. Embed SL(k) in SL(n). Let us consider two examples of the many possible embeddings.

1) top left corner: ( \begin{matrix}SL(k) & 0 \\ 0 & Id \end{matrix} )

2) embed SL(k) ↪ SL(n) via an irreducible representation.

Even these simple cases are not completely understood. Ergodic theory can be applied to study these to cases, but completely different ergodictheoretic ideas have been used for the two different cases.

1) Labourie, Mozes, and Zimmer used superrigidity arguments in a series of papers.

2) Margulis and Oh used a nonergodicity theorem based on the rate of decay of matrix coefficients along certain subgroups.

The techniques are complementary, in that Ergodic theory, groups, and geometry Informal lecture notes — not for publication

the method used in (2) does not apply to any examples of type (1).

Labourie and T. Kobayashi have written surveys on related work.

How to use superrigidity. As an example, con-

sider the case G = SL(n), H = ( \begin{matrix}SL(k) & 0 \\ 0 & Id \end{matrix} ),

and assume n − k ≥ 3.

Suppose Γ\G/H is a compact lattice form, and let L ≅ SL(n − k) be a subgroup of the centralizer Z_{G}(H).

Observation. L acts on M (because L centralizes H), preserving a finite measure (because Γ and H are unimodular). Furthermore, Γ\G is a principal H-bundle over M, and R-rank(L) ≥ 2 (because n − k ≥ 3), so we are in the setting to apply superrigidity.

For simplicity, assume k < n/2. (This simplifies the proof, because, in this case, L is bigger than H, so there is no nontrivial homomorphism L → H. The argument can be extended to eliminate this assumption.)

Thus, the homomorphism π: L → H in the conclusion of the cocycle superrigidity theorem is trivial, so there is an L-invariant reduction of the principal bundle Γ\G to a compact subgroup C ⊂ H. That is, there is an L-invariant section s: M → Γ\G/C. Then, by taking a C-invariant metric on the fibers, and considering the characteristic function of the set of all points in the fiber within distance 1 of the section, we see that there is an L-invariant L^{2} function on Γ\G.

So the natural unitary representation of G on L^{2}(Γ\G) has an L-invariant vector. Because L is noncompact, the Moore Ergodicity Theorem implies that there is a G-invariant vector in L^{2}(Γ\G), so Γ is a lattice in G. This is impossible, because H = SL(n − k) is not compact.

Thm (Labourie, Mozes, Zimmer). Embed H = SL(k) in G = SL(n) in the top left corner, and assume n ≥ 6. Then G/H has:

• no compact forms if k ≤ n/2;

• no compact lattice forms if k ≤ n − 3.

Benoist and Margulis (independently) showed that SL(n)/ SL(n − 1) has no compact lattice form if n is odd. It seems that the case SL(3)/ SL(2) is still open for non-lattice forms, and that the cases SL(4)/ SL(3) and SL(n)/ SL(n − 2) are open even for lattice forms.

The approach of Margulis and Oh.

Defn. Let (π, H) be a unitary representation of G. For any nonzero vectors v, w ∈ H, the function G → C defined by g ↦ ⟨π(g)v | w⟩ is a matrix coefficient of π. For simplicity, we always assume v and w are K-invariant, where K is a fixed maximal compact subgroup of G.

When G is a simple group with finite center, and π has no invariant vectors, each matrix coefficient tends to 0 as g → ∞. The rate of decay to 0 is an important subject of study.

Defn. Fix a simple Lie group G with finite center, and a maximal compact subgroup K. We say a subgroup H is tempered in G if ∃q ∈ L^{1}(H), such that for every unitary representation (π, H) of G with no invariant vectors, and all K-invariant vectors v, w ∈ H, we have

|⟨π(h) | w⟩| ≤ q(h) ||v|| ||w||.

Some subgroups of G are tempered and some are not. (G is not a tempered subgroup of itself.) Hee Oh proved a theorem that, in many cases, tells us whether or not a subgroup is tempered.

Eg. (Oh). 1) Embed SL(2) ↪ SL(n) via an irreducible representation. The image is a tempered subgroup of SL(n) iff n ≥ 4.

2) The subgroup SL(k), embedded in the top left corner of SL(n), is not tempered.

Vanishing of matrix coefficients implies an ergodicity theorem (namely, the Moore Ergodicity Theorem), but Margulis used the tempered condition to prove a nonergodicity theorem:

Thm (Margulis). Suppose G acts on a measure space X with an invariant measure that is σ-finite but not finite. If H is a tempered subgroup of G, then H is not ergodic on X.

Here is a more precise topological statement of the result:

Thm (Margulis). Suppose G acts on a noncompact, locally compact space X, preserving a σ-finite Radon measure. If H is a tempered subgroup of G and A is a compact subset of X, then HA ≠ X.

Note that the conclusion of the topological version clearly implies that H is not ergodic on X.

Cor. If H is a tempered, noncompact subgroup of G, then G/H has no compact lattice form.

Proof. If Λ\G/H is compact, then there is a compact set A ⊂ Λ\G with AH = Λ\G. This contradicts the conclusion of the theorem.

9. Stationary measures & projective factors.

Suppose G acts on X.

Previous lectures have concentrated on the case where there is a finite G-invariant measure on X. We now discuss what to do when there is no finite invariant measure. In particular, we want a good substitute for a finite invariant measure. If X is a manifold, one could take a smooth measure on X. In general, one could take a measure on an orbit, but such a measure often is not a good reflection of the space. For example, compact sets may not have finite measure.

Eg. Let Q be a parabolic subgroup of G, and let M = G/Q. Then G acts on M, with no finite invariant measure (unless Q = G).

Question. How close is a general action to being a combination of two basic types:

1) an action with finite invariant measure; and

2) the action on some G/Q?

Let μ be a probability measure on G. Assume • μ is symmetric;

• μ is absolutely continuous (w.r.t. Haar); and

• supp(μ) generates G. We then say that μ is “admissible.”

Given such a measure μ on G and a measure ν on X, define a measure μ ∗ ν on X by

μ ∗ ν = ∫_{G} g_{∗}ν dμ(g);

i.e., (μ ∗ ν)(A) = ∫_{G} ν(g^{−1}A) dμ(g).

Defn. ν is called μ-stationary (or simply stationary) if μ ∗ ν = ν.

Basic properties. 1) If X is compact, then a stationary measure always exists.

2) Stationary measures are quasi-invariant.

3) Suppose K is a compact subgroup of G, Ergodic theory, groups, and geometry Informal lecture notes — not for publication

the measure μ is K-invariant, and ν is stationary. Then ν is K-invariant.

Proof. 1) ν ↦ μ∗ν is an affine map on the space of probability measures on X. The space of probability measures is a convex compact set, so a standard fixed-point theorem implies there is a stationary measure.

(2) requires argument, but uses standard techniques.

(3) is easy.

Eg. Suppose G is simple and X = G/Q for a parabolic subgroup Q. If μ is K-invariant, then, since K is transitive on G/Q, the basic properties imply that the stationary measure is the unique K-invariant probability measure on G/Q.

If we change μ, the set of μ-stationary measures will change, but the set of measure classes will not change. (We are still assuming G is a simple Lie group.) Thus, there is a natural collection of measures (“natural” meaning that it depends only on G, not on the choice of μ), namely, the measure classes that contain a stationary measure. Unfortunately, it may not be obvious which measure classes are members of the collection.

Eg. Suppose Q is a parabolic subgroup of G, and suppose Q acts on a compact space Y with finite invariant measure λ.

Induce to G: let X = (G × Y)/Q. Then X is a bundle over G/Q with fiber Y.

Because the base and fiber are compact, we know that X is compact.

By combining the stationary measure ν_{G/Q} on the base with the invariant measure λ on the fibers, we obtain a quasi-invariant measure on X. This measure is stationary.

Note that, in this example, the fiber measures are invariant (that is, the maps from fiber to fiber are measure preserving) — all the non-invariance is happening on the base G/Q.

Rem. Suppose a semisimple group G acts on a compact space X. Let P be a minimal parabolic subgroup of G, and let K be a maximal compact subgroup of G.

Because P is amenable, there is a P -invariant probability measure λ on X. Then ν = Haar_{K} ∗ λ is μ-stationary, for any K-invariant admissible measure μ. (This works because Haar_{K} projects to the stationary measure on G/P.)

Thm (Furstenberg). If G is a simple Lie group, and μ is K-invariant and admissible, then the map λ ↦ Haar_{K} ∗ λ is an affine bijection between the space of P -invariant probability measures on X and the (G, μ)-stationary measures on X.

Thus, every stationary measure can be obtained by starting with a P -invariant measure, and averaging over K.

We mentioned above that the collection of stationary measures depends on the choice of μ. Furstenberg’s theorem shows that, in the case where μ is K-invariant, it does not depend on the choice of the specific measure μ, but only the choice of the maximal compact subgroup K.

Problem. 1) Find conditions under which any X with a stationary measure is (up to measurable isomorphism, or up to C^{∞} isomorphism) induced from a measure-preserving action of a parabolic.

2) For R-rank(G) ≥ 2, does every (X, ν) have some G/Q as a quotient (unless ν is invariant)? (We assume ν is stationary.) In other words, is (X, ν) induced from some action (not necessarily measure preserving) of some parabolic subgroup?

Rem. 1) For any G, there are actions not of the type described in (1). (2) is open, but here is a counterexample in rank one.

Counterexample to (2) in real rank one. Let Γ Ergodic theory, groups, and geometry Informal lecture notes — not for publication

be a cocompact lattice in SL(2, R), such that Γ has a homomorphism onto the free group F_{3} on three generators. Then F_{3} maps onto F_{2}, and F_{2} embeds in SL(2, R) as a lattice. The composition is a homomorphism from Γ into SL(2, R) with a large kernel. By restricting the usual projective action of SL(2, R) on S^{1}, we obtain an action of Γ on S^{1}.

Induce the Γ-action to an action of SL(2, R): let X = (SL(2, R) × S^{1})/Γ. Then X is a circle bundle over SL(2, R)/Γ.

Claim. There is no measurable G-map X → G/P, where P is a proper parabolic subgroup of G. If there were such a G-map, then, because the Gaction on X is induced from the Γ-action on S^{1}, there is a Γ-map f: S^{1} → G/P. Let N be the kernel of the composite homomorphism Γ → F_{2}, so N acts trivially on S^{1}. Then N acts trivially on G/P a.e. (w.r.t. f_{∗}ν, where ν is the quasiinvariant measure on S^{1}), so N fixes some point in G/P. Therefore, N is contained in a conjugate of P. Because P is solvable, we conclude that N is solvable. This contradicts the fact that infinite normal subgroups of lattices cannot be solvable (because, by the Borel Density Theorem, a normal subgroup of a lattice is Zariski dense in G). Defn. Suppose G acts on X with stationary measure ν. Let λ be the corresponding P -invariant measure (Furstenberg’s theorem). We say the Gaction on X is P -mixing if the action of P on (X, λ) is mixing, i.e., if λ(h_{n}A ∩ B) → λ(A) λ(B) as h_{n} → ∞ in P. Thm (Nevo-Zimmer). Suppose G acts on a compact space X with stationary measure ν. Assume

(1) the action is P -mixing and (2) R-rank(G) ≥ 2.

Then there is a parabolic subgroup Q and a Qspace Y with finite invariant measure, such that X is the induced G-space.

Rem. The conclusion can fail if either (1) or (2) is omitted.

Defn (Furstenberg entropy). Suppose G acts on X, with a μ-stationary measure ν. Define

h(ν) = − ∫_{X×G} log \frac{dg_{∗}ν}{dν} (x) dμ dν.

Facts. 1) h(ν) ≥ 0.

2) h(ν) = 0 iff ν is G-invariant.

3) If f: (X_{1}, ν_{1}) → (X_{2}, ν_{2}) is G-equivariant, and v_{1} and ν_{2} are stationary, then h(ν_{1}) ≥ h(ν_{2}). Furthermore, h(ν_{1}) = h(ν_{2}) iff there is a “relative invariant measure.”

The only case we need of (3) is that if X is induced from a finite measure-preserving action of a parabolic subgroup Q, then h(X) = h(G/Q).

Cor. Assume R-rank(G) ≥ 2. The set of Furstenberg entropies h(X) of P -mixing actions of G is a finite set, namely

{ h(G/Q) | Q is parabolic }.

The corollary is false in real rank one, even with the P -mixing assumption.

If Q_{1} ⊂ Q_{2}, then there is a G-map from G/Q_{1} to G/Q_{2}, so h(G/Q_{1}) > h(G/Q_{2}). Thus, we have the following corollary.

Cor. Assume R-rank(G) ≥ 2. Then ∃c > 0 such that, for every P -mixing action of G on X, if h(X) < c, then there is a Q-invariant measure for some maximal parabolic subgroup Q.

Eg. Let G = SL(n), Then a maximal parabolic is

of the form Q = ( \begin{matrix}∗ & ∗ \\ 0 & ∗ \end{matrix} ), so Q must contain

a copy of SL(n/2). Thus we can apply all of our theory to the action of SL(n/2).

For example, suppose G = SL(n) acts on X with a stationary measure and the action is engaging. If h(X) ≤ c, then π_{1}(X) contains an arithmetic subgroup of a group L, where L ⊃ SL(n/2).

For a larger and larger values of c, we obtain weaker and weaker conclusions, until we reach the point at which we obtain no conclusion.

Upshot. If the Furstenberg entropy is small, then we can apply the entire invariant measure program to the action of a group L ⊂ Q with R-rank(L) ≥ 2.

10. Orbit equivalence

In this talk, we discuss some exciting developments in the last year.

Assumption. All actions are assumed to be ergodic and non-transitive, with a finite invariant measure.

Notation. If G acts on X, we use Rel(X, G) to denote the equivalence relation on X defined by the G-orbits, so Rel(X, G) ⊂ X × X.

Defn. Suppose (G_{1}, X_{1}) and (G_{2}, X_{2}) are actions. We say the actions are orbit equivalent if there is a measure-preserving bijection φ: X_{1} → X_{2} with φ(G_{1}-orbit) = G_{1}-orbit, i.e., if Rel(X_{1}, G_{1}) ≅ Rel(X_{2}, G_{2}).

Defn. The actions (G_{1}, X_{1}) and (G_{2}, X_{2}) are stably orbit equivalent if Rel(X_{1} × I, G_{1} × I) ≅ Rel(X_{2} × I, G_{2} × I), where I is the closed unit interval [0, 1].

Technical facts. 1) If the G_{1} and G_{2} orbits are a.e. not countable, then stable orbit equivalence is the same as orbit equivalence.

2) If G_{1} and G_{2} are discrete, then the actions (G_{1}, X_{1}) and (G_{2}, X_{2}) are stably orbit equivalent iff there are subsets A_{1} ⊂ X_{1} and A_{2} ⊂ X_{2} of positive measure, such that Rel(X_{1}, G_{1})|_{A_{1}} ≅ Rel(X_{2}, G_{2})|_{A_{2}}. (That is, after restricting to two sets of positive measure, we have an orbit equivalence.)

3) If a discrete subgroup Γ of G acts on X, then the Γ-action on X is stably orbit equivalent to the induced G-action.

A classical example similar to (3) is the construction of a flow built under a function. It is clear that the orbits of the flow are the same as the orbits of the discrete dynamical system, but with the points extended into intervals.

Thm (Dye). Any two Z-actions are orbit equivalent.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

Thm (Ornstein-Weiss). If G_{1} and G_{2} are discrete amenable groups, then any action of G_{1} is orbit equivalent to any action of G_{2}.

Thus, in the amenable case, the orbit structure tells you nothing about the groups, and nothing about the actions. The following theorem shows that the situation for simple groups is exactly the opposite.

Thm (Zimmer). Suppose G_{1} and G_{2} are noncompact, simple Lie groups with R-rank(G_{i}) ≥ 2. If (G_{1}, X_{1}) is orbit equivalent to (G_{2}, X_{2}), then G_{1} is locally isomorphic to G_{2}, and, up to a group automorphism, the actions are isomorphic.

Defn (cocycle defined by an orbit equivalence). Suppose φ: X_{1} → X_{2} with φ(G_{1}-orbit) = G_{2}-orbit. For any g ∈ G and x ∈ X_{1}, the points φ(gx) and φ(x) are in the same G_{2}-orbit, so there is some c(g, x) ∈ G_{2} with

φ(gx) = c(g, x) · φ(x). The resulting map c: G_{1} × X_{1} → G_{2} is a cocycle, i.e., for all g, h ∈ G_{1} and a.e. x ∈ X_{1}, we have c(gh, x) = c(g, hx) c(h, x).

In the situation of the theorem, we may apply the Cocycle Superrigidity Theorem to conclude that the cocycle c is cohomologous to a homomorphism, that is, a cocyle that is independent of x. This yields a (local) isomorphism from G_{1} to G_{2} and, after identifying G_{1} with G_{2} under this isomorphism, the actions are isomorphic. This proves the theorem.

Question. What about real rank one?

There are some results of Cowling-Zimmer on Sp(1, n), but the general case is open.

Generally, the superrigidity arguments in real rank one use some kind of integrability condition on the cocycle, for example, that, for fixed g, the function c(g, x) is an L^{1} function of x. But the cocycle obtained from an orbit equivalence typically has no regularity, beyond being measurable, so the arguments do not apply.

Question. What about actions of lattices?

Prop. If Γ and Λ are lattices in G, then the Γaction on G/Λ is stably orbit equivalent to the Λaction on G/Γ.

Proof. Induce to G: both yield the G-action on G/Γ × G/Λ. The induced actions are isomorphic, hence stably orbit equivalent, so the original actions are stably orbit equivalent.

In particular, free groups F_{m} and F_{n} have stably orbit equivalent actions, because both are lattices in SL(2, R).

Prop. Let Γ and Λ be lattices in a simple Lie group G of higher rank. The Γ-action on G/Λ is orbit equivalent to the Λ-action on G/Γ (not just stably orbit equivalent) iff vol(G/Γ) = vol(G/Λ).

All this was known 15 years ago, and not much changed until this year, with the results of Furman and Gaboriau.

Thm (Furman). Suppose Γ is a lattice in a simple Lie group G of higher rank, and Γ acts on (X, μ). For every lattice Δ in G, assume there is no measure-preserving Γ-equivariant map X → G/Δ.

Let any discrete group Λ act on a space Y. If (Γ, X) is orbit equivalent to (Λ, Y), then Γ ≅ Λ, and the actions are isomorphic.

Note that Λ is not assumed to be a lattice in Furman’s theorem. It is not even assumed to be a linear group — it can be any group.

Witte classified the quotients of standard actions, from which it is often obvious that there is no quotient of the form G/Δ, so we have the following corollary.

Ergodic theory, groups, and geometry Informal lecture notes — not for publication

Cor. The conclusion of the theorem is true for: • the natural action of SL(n, Z) on T^{n};

• the action of Γ on K/H, arising from an embedding of Γ in a compact group K, and H is a closed subgroup;

• The action of Γ on H/Δ arising from an embedding of G as a proper subgroup of a simple group H, and Δ is a lattice in H.

Gaboriau’s work. These results have been announced, but there is no preprint yet.

Defn. Let R be a countable equivalence relation on X (i.e., each equivalence class is countable), and assume μ(X) = 1.

Suppose Φ is a countable collection of maps φ_{i}: A_{i} → B_{i}, where A_{i}, B_{i} ⊂ X, such that φ_{i}(x) ∼ x. Assume {A_{i}} covers X.

Let R_{Φ} be the equivalence relation generated by Φ. Then we say that Φ generates R if R_{Φ} = R.

Define the cost c(Φ) = ∑_{i} μ(A_{i}). Then 1 ≤ c(Φ) ≤ #(Φ).

Defn. c(R) = inf{ c(Φ) | Φ generates R }.

Prop. If R = R(X, Γ) and Γ is amenable, then c(R) = 1.

Proof. By Ornstein-Weiss, (X, Γ) is orbit equivalent to a Z-action. Thus, the equivalence relation is generated by the action of a single transformation, so we may take Φ to be a one-element set.

Observe that Φ defines the structure of a graph on each equivalence class. Gaboriau gives a condition, in terms of these graphs, for when a particular set Φ of transformations attains the infimum.

Thm (Gaboriau). If the graph resulting from Φ on each equivalence class is a tree (so, in particular, the equivalence relation is “treeable” in the sense of S. Adams), then c(R) = c(Φ).

Cor. Let F_{n} be the free group on n generators.

For a free action of F_{n} on any space X, we have c(Rel(X, F_{n})) = n.

Cor. Free groups F_{m} and F_{n} do not have free orbit-equivalent actions (unless m = n).

Rem. c(Rel(X, Γ)) = 1 for any action of a lattice Γ in a simple group of higher rank.

Relationship to quasi-isometries.

Defn. Let (X, d_{1}) and (Y, d_{2}) be locally compact metric spaces. A quasi-isometry is a map φ: X → Y (not necessarily continuous), such that there are positive constants C and K such that

\frac{1}{C} d(x, y)−K ≤ d(φ(x), φ(y)) ≤ Cd(x, y)+K

and φ(X) is K-dense in Y (i.e., the K-ball around φ(X) is all of Y).

Eg. 1) Let M be a compact manifold. For any two metrics ω_{1} and ω_{2} on M, the identity map is a quasi-isometry from (M̃, ω̃_{1}) to (M̃, ω̃_{2}), so M̃ has a canonical metric up to quasi-isometry.

2) Let Γ be a finitely generated group. For each finite generating set F, the group Γ can be made into a metric space by identifying it with the vertices of the Cayley graph Cay(Γ; F). The resulting metric is not canonical, because it depends on the generating set, but the metric is canonical up to quasi-isometry.

Prop. If M is a compact manifold, then the fundamental group π_{1}(M) is quasi-isometric to the universal cover M̃.

In particular, any two cocompact lattices in the same simple Lie group G are quasi-isometric, because they are fundamental groups of manifolds with the same universal cover, namely the symmetric space associated to G.

Thm (Schwartz, Farb, Eskin). For non-cocompact lattices, quasi-isometry is the same as commensurability. Furthermore, any finitely generated group quasi-isometric to a non-cocompact lattice is virtually isomorphic to it.

In particular, non-cocompact lattices in two different simple Lie groups are not quasi-isometric. Quasi-isometric non-cocompact lattices in the same group are commensurable.

Possible connection.

• Quasi-isometry: Γ and Λ are quasi-isometric iff there is an action of Γ×Λ on a locally compact space X such that each of Γ and Λ acts properly discontinuously and cocompactly.

• Stable orbit equivalence: Γ and Λ have stably orbit equivalent actions iff there is an action of Γ×Λ on a measure space Y with σ-finite invariant measure μ, such that each of Γ and Λ acts properly with quotient of finite measure (i.e., Y/Γ and Y/Λ have finite measure).

If one could find a σ-finite invariant measure on X (whose quotients by Γ and Λ are finite), there would be a direct connection between quasiisometry and stable orbit equivalence.

Problem. Understand the measure theory of quasi-isometries.

11. Appendix to Lecture 5:

ideas of Gromov’s proof

Assume G acts on M, preserving a connection ω.

Problem. Construct V ⊂ Vect(M̃) such that

1) V is finite-dimensional and “large”: ∀x ∈ M̃, ev_{x}(V) ⊃ ev_{x}(G) = T (G-orbits).

2) V centralizes G.

3) V is Γ-invariant.

We need a machine to generate new vectorfields on M̃ preserving ω.

Let P^{(k)} be the kth-order frame bundle over M (with fiber GL^{(k)}).

The kth-order tangent bundle T^{(k)}M is an associated vector bundle for P^{(k)}; the fiber T^{(k)}M_{x} is the space of (k − 1)-jets of vector fields at x.

For each x ∈ M, we have G_{x}^{(k)} ⊂ T^{(k)}M_{x}, where G_{x}^{(k)} is the space of (k − 1)-jets of elements of the Lie algebra G, considered as a subspace of Vect(M).

Let L^{(k)} be the algebraic hull of the G-action on P^{(k)}, the “kth-order algebraic hull,” so L^{(k)} ⊂ GL^{(k)}.

In a previous lecture, we showed that G ↪ L = L^{(1)}, and the homomorphism G → L contains Ad. We use the same argument but in kth order instead of first order.

We have G^{(k)} ⊂ T^{(k)}M_{x}, and the G-action on G^{(k)} is the adjoint action, so the Borel Density Theorem implies that L^{(k)} ⊃ G, and, in the concrete representation of L^{(k)} on T^{(k)}M, the representation of G contains Ad.

Because G also preserves the connection ω, we will be able to say more about L^{(k)}.

The connection ω is a reduction of a GL^{(2)}-bundle to GL^{(1)}. Since GL^{(2)} acts on connections at a Ergodic theory, groups, and geometry Informal lecture notes — not for publication

point, the group GL^{(k)} acts on (k − 2)-jets of connections at a point.

Because G preserves the connection ω, it preserves a section of the bundle whose fiber at each point is the space of (k − 2)-jets of connections.

Therefore, for each x ∈ M, the realization of L^{(k)} is contained in the group fixing a (k − 2)- jet of ω_{x}. (This is analogous to our conclusion in the case of actions on Lorentz manifolds that L ⊂ SO(1, m).)

Notation. Let Aut^{(k)}(M, ω; x) be the set of k-jets of maps at x that preserve ω up to order k.

The discussion so far tells us that

GL^{(k)} ⊃ Aut^{(k−2)}(M, ω; x) ⊃ L^{(k)} ⊃ Ad_{G^{(k)}} G Ignore L^{(k)}: we have

Aut^{(k−2)}(M, ω; x) ⊃ Ad_{G^{(k)}} G, which means that, at each point, there are enough automorphisms up to order k to contain all of Ad(G).

Three steps remain:

Step 1. Pass from Aut^{(k−2)}(M, ω; x) to Aut^{(loc)}(M, ω; x). That is, take the infinitesimal automorphisms and turn them into local automorphisms. This is achieved by using the Frobenius Theorem (“rigid properties of connections”); passing from infinitesimal automorphisms to local automorphisms is a well-developed topic in differential geometry.

Step 2. Pass from local automorphisms on M̃ to global automorphisms. This is based on analytic continuation. For vector fields preserving connections, this type of result appears in Nomizu.

Steps 1 and 2 show that there exist many new vector fields on M̃ preserving ω.

• By “many,” we mean that the vector fields normalize G, and the representation on G contains Ad G.

• By “new,” we mean we have many that leave x fixed (or, we should say that the vector fields vanish at x).

Step 3. Go from normalizing to centralizing. This step uses the simplicity of G: an element of the normalizer differ from the centralizer only by an element of G.

A lot more work is needed to fill in all the details of the argument. It would be a great service for someone to write a good and complete exposition of the proof.