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BOREL SUPERRIGIDITY OF SL2(O)-ACTIONS

SCOTT SCHNEIDER

Abstract. We prove a Borel superrigidity theorem for suitably chosen actions of groups

of the form SL2(O), where O is the ring of integers inside a multi-quadratic number field

Q(√p1, . . . ,

√ps), and present applications to the theory of countable Borel equivalence

relations. In particular, for suitable primes p 6= q we prove that the orbit equivalence

relations arising from the natural actions of SL2(Z[√q]) on the p-adic projective lines are

incomparable with respect to Borel reducibility as p, q vary.

1. Introduction

Let p 6= q be odd primes and let PG(1,Qp) = Qp ∪∞ be the projective line over

the field Qp of p-adic numbers. If q is a quadratic residue modulo p, then q has a

square root in Qp, and hence SL2(Z[√q]) has a natural Borel action on PG(1,Qp) as

a group of fractional linear transformations; i.e., a b

c d

· x =ax+ b

cx+ d

for each x ∈ PG(1,Qp). Let Epq be the resulting orbit equivalence relation. In this

paper we will prove a Borel superrigidity theorem for actions of this form, and show

in particular that if p′ 6= p is a second prime such that√q ∈ Qp′ , then Epq and Ep

′q

are incomparable with respect to Borel reducibility.

In [18] Thomas proved an analogous result for actions of groups of the form

SL2(Z[1q ]) on the p-adic projective lines. This in turn extended similar results for

actions of SLn(Z) on higher dimensional projective spaces obtained in [17]. Specifi-

cally, in [17] Thomas proved the following:

Theorem 1.1 (Thomas [17, 6.7]). Suppose n ≥ 3, and for each prime p let Ep be the

orbit equivalence relation arising from the natural action of SLn(Z) on PG(n−1,Qp).

If p 6= q, then Ep and Eq are incomparable with respect to Borel reducibility.1

2 SCOTT SCHNEIDER

Theorem 1.2 (Thomas [17, 5.1]). Suppose n ≥ 3, and let J1 and J2 be nonempty

sets of primes. For i = 1, 2, denote by EJi the orbit equivalence relation arising from

the translation action (as a subgroup via the diagonal embedding) of SLn(Z) on

K(Ji) =∏p∈Ji

SLn(Zp),

where Zp is the ring of p-adic integers. If J1 6= J2, then EJ1 and EJ2 are incomparable

with respect to Borel reducibility.

The proofs of these theorems depended essentially upon Zimmer’s measure theo-

retic superrigidity results [22, 5.2.5 and 10.1.6] for irreducible lattices in higher rank

semisimple Lie groups. Unfortunately, Zimmer’s theorem fails for the low rank Lie

group SL2(R), and hence there is no hope of applying the ideas of Thomas [17] to

the orbit equivalence relations arising from the analogous actions of SL2(Z) on the

projective lines PG(1,Qp). This is especially unfortunate considering that analogues

of the above theorems for n = 2 would yield infinitely many treeable countable Borel

equivalence relations that are incomparable with respect to Borel reducibility, thus

solving a major open problem in the field of countable Borel equivalence relations

(see, for instance, [8, Section 3]).

Indeed, Thomas conjectured ([17, 5.7], [17, 6.10]) that Theorems 1.1 and 1.2 should

still hold for n = 2. With this context in mind, Thomas’ subsequent results in [18]

may be viewed as attempts to extend Theorems 1.1 and 1.2 as far as possible in the

direction of Conjectures [17, 5.7] and [17, 6.10], while still appealing to Zimmer super-

rigidity. Specifically, in [18] Thomas proved the following analogues of Theorems 1.1

and 1.2, making essential use of the fact that groups of the form

∆S = SL2(Z[1/p1, . . . , 1/ps]),

S = p1, . . . , ps a finite set of rational primes, may be realized as irreducible lattices

in the higher rank semisimple Lie groups

SL2(R)× SL2(Qp1)× · · · × SL2(Qps).

BOREL SUPERRIGIDITY OF SL2(O)-ACTIONS 3

Theorem 1.3 (Thomas [18, 1.1]). Suppose that p, q are primes and that S, T are

finite nonempty sets of primes such that p 6∈ S and q 6∈ T . Let EpS be the orbit

equivalence relation arising from the action of ∆S on PG(1,Qp). If (p, S) 6= (q, T ),

then EpS and EqT are incomparable with respect to Borel reducibility.

Theorem 1.4 (Thomas [18, 1.2]). Suppose that S1 and S2 are finite nonempty sets

of primes and that J1, J2 are (possibly infinite) nonempty sets of primes such that

S1 ∩ J1 = S2 ∩ J2 = ∅. For i = 1, 2, let EJiSi

be the orbit equivalence relation arising

from the translation action (as a subgroup via the diagonal embedding) of ∆Si on

K(Ji) =∏p∈Ji

SL2(Zp),

where Zp is the ring of p-adic integers. If (J1, S1) 6= (J2, S2), then EJ1S1

and EJ2S2

are

incomparable with respect to Borel reducibility.

The present work may be viewed as a continuation of the study initiated in [18], i.e.,

as another attempt to extend Theorems 1.1 and 1.2 in the direction of Conjectures

[17, 5.7] and [17, 6.10] while still relying upon Zimmer superrigidity. We will consider

actions of lattices of the form

ΓS = SL2(OS),

where again S = p1, . . . , ps is a finite nonempty set of rational primes, and where

OS is the ring of integers inside the algebraic number field

Q(√p1, . . . ,

√ps).

Thus we shall obtain the following analogues of Theorems 1.3 and 1.4.

Theorem 1.5. For i = 1, 2, let Si be a finite, nonempty set of primes, pi 6∈ Si a

prime such that√q ∈ Qpi for all q ∈ Si, so that ΓSi acts on PG(1,Qpi) as a group of

fractional linear transformations. Then writing Epi

Sifor the orbit equivalence relation

arising from this action, we have that Ep1S1and Ep2S2

are incomparable with respect to

Borel reducibility whenever (p1, S1) 6= (p2, S2).

4 SCOTT SCHNEIDER

Theorem 1.6. For i = 1, 2, let Si and Ji be nonempty sets of primes, Si finite,

Si∩Ji = ∅. Suppose that√p ∈ Zq for all p ∈ Si and q ∈ Ji, so that ΓSi is a subgroup of

K(Ji) =∏p∈Ji

SL2(Zp) via the diagonal embedding. Let EJiSi

be the orbit equivalence

relation arising from the left translation action of ΓSi on K(Ji). Then EJ1S1

and EJ2S2

are incomparable with respect to Borel reducibility whenever (S1, J1) 6= (S2, J2).

These results will be shown to follow from a more general Borel superrigidity the-

orem, which we state below as Theorem 3.1. The majority of our effort shall go into

proving 3.1.

It should be pointed out that the techniques used to prove Theorems 1.1 and 1.2

in Thomas [17] rely upon the fact that for n ≥ 3, SLn(Z) is Kazhdan. However, the

groups ∆S are not Kazhdan, and hence some modification to the techniques of [17] was

required in order to prove Theorems 1.3 and 1.4 in Thomas [18]. Fortunately, while

the groups ∆S are not Kazhdan, they do possess the weaker Property (τ), and indeed

it was the central insight of Thomas [18] to realize that in the context of Theorems

1.3 and 1.4, this weaker property suffices to push through the arguments used in [17].

As the groups ΓS also possess Property (τ) but are not Kazhdan, in proving Theorem

3.1 we will make essential use of the technology developed in Thomas [18].

This paper is organized as follows. In Section 2, we shall recall some basic notions

and results concerning Borel equivalence relations, ergodic theory, Borel cocycles,

and irreducible lattices in semisimple Lie groups. In Section 3, we state our Borel

superrigidity theorem, 3.1, and prove from it Theorems 1.5 and 1.6. Then in Section

4 we will discuss some of the main ideas appearing in the proof of 3.1 and sketch a

brief outline of our argument, indicating along the way some of the key lemmas we

shall need later in the body of the proof. In Section 5 we formally state and prove

these lemmas, and then in Section 6 we prove Theorem 3.1.

Acknowledgements. This paper would not have been possible without the dedi-

cated assistance of my thesis advisor, Simon Thomas. Additionally I would like to

thank Samuel Coskey and Richard Lyons for some helpful discussions.


2. Preliminaries

In this section we recall some basic notions and results concerning Borel equivalence

relations, ergodic theory, Borel cocycles, and irreducible lattices in semisimple Lie

groups.

2.1. Borel equivalence relations. Let X be a standard Borel space, i.e., a Polish

space equipped with its σ-algebra of Borel sets. A Borel equivalence relation on X

is an equivalence relation E ⊆ X2 that is Borel as a subset of X2; E is said to be

countable iff each E-class is countable.

If E and F are Borel equivalence relations on the standard Borel spaces X and Y ,

respectively, then a Borel homomorphism from E to F is a Borel function f : X → Y

such that xEy implies f(x)Ff(y) for all x, y ∈ X. A Borel homomorphism for which

xEy iff f(x)Ff(y) is called a Borel reduction. If there is a Borel reduction from E to

F then we say that E is Borel reducible to F , and write E ≤B F . We say that E and

F are Borel bireducible, and write E ∼B F , iff both E ≤B F and F ≤B E.

The Borel equivalence relations we shall consider in this paper arise from group

actions as follows. Let G be a lcsc group, i.e., a locally compact second countable

group. Then a standard Borel G-space is a standard Borel space X together with a

Borel action (g, x) 7→ g ·x of G on X. The corresponding G-orbit equivalence relation

on X, denoted EXG , is a Borel equivalence relation which by Kechris [9] is bireducible

with a countable Borel equivalence relation. Conversely, by Feldman-Moore [5], if E is

an arbitrary countable Borel equivalence relation on the standard Borel space X, then

there exists a countable group G and a Borel action of G on X such that E = EXG . For

a detailed development of the general theory of countable Borel equivalence relations

we direct the reader to Jackson-Kechris-Louveau [8].

2.2. Ergodic theory. Suppose that G is a lcsc group and X is a standard Borel

G-space. A Borel measure µ on X is called G-invariant iff µ(A) = µ(gA) for all

g ∈ G and Borel A ⊆ X. (In this paper, all measures on Polish spaces will be Borel

measures, i.e., measures defined on the σ-algebra of Borel sets). A standard Borel

6 SCOTT SCHNEIDER

G-space (X,G, µ) with invariant Borel measure µ will sometimes be called a standard

Borel system.

If µ is a G-invariant measure on X, then the action of G on X is said to be

ergodic iff every G-invariant Borel subset of X is µ-null or µ-conull. The following

characterization of ergodicity is well-known.

Proposition 2.1. Let G be a lcsc group, X a standard Borel G-space with G-

invariant Borel measure µ. Then the following are equivalent:

(1) the action of G on (X,µ) is ergodic;

(2) for any standard Borel space Y and G-invariant Borel function f : X → Y , f

is µ-a.e. constant.

(3) every G-invariant Borel function f : X → [0, 1] is µ-a.e. constant.

If E is a countable Borel equivalence relation on a standard Borel space X and µ

is a Borel measure on X, then µ is said to be E-invariant iff µ is H-invariant for

some (equivalently every) countable group H and Borel action of H on X such that

E = EXH . If E, F are Borel equivalence relations on the standard Borel spaces X, Y ,

respectively, and µ is an E-invariant Borel measure on X, then (E,µ) (or simply E,

if µ is understood) is said to be F -ergodic iff for any Borel homomorphism f : X → Y

from E to F , there exists a µ-conull subset of X that f maps into a single F -class.

It is easily checked that if F ∼B F ′, then E is F -ergodic iff E is F ′-ergodic, and that

if E is F -ergodic and F ′ ≤B F , then E is F ′-ergodic. In light of 2.1, ergodicity is

equivalent to ∆(Y )-ergodicity, where ∆(Y ) is the identity relation on any uncountable

standard Borel space Y .

Throughout this paper, E0 will denote the Vitali equivalence relation defined on

2N by αE0 β iff α(n) = β(n) for all but finitely many n. If X is a standard Borel

G-space with invariant Borel measure µ and EXG is E0-ergodic, then the action of G

on X is easily seen to be ergodic.

The action of G on X is said to be uniquely ergodic iff there exists a unique G-

invariant probability measure µ on X. In this case it is well-known that µ must be

ergodic. (For example, see Bekka-Mayer [4, 4, Sec. I.3]).


If Γ is a countable group, X a standard Borel Γ-space with invariant ergodic prob-

ability measure µ, and Λ ≤ Γ a finite-index subgroup, then a positive measure,

Λ-invariant Borel set Z ⊆ X is called an ergodic component for the action of Γ on X

iff Λ acts ergodically on (Z, µZ), where µZ is the probability measure defined on Z by

µZ(A) = µ(A)/µ(Z) for A ⊆ Z Borel. In this case it is well known that there exists

a partition X = Z1 t · · · t Zd of X into finitely many ergodic components, uniquely

determined up to null sets, and that the action of Λ on each component is uniquely

ergodic if the Γ-action on X is uniquely ergodic.

Finally, if X and Y are standard Borel spaces, f : X → Y is a Borel function, and

µ is a probability measure on X, then the measure f∗µ defined on Y by

(f∗µ)(A) = µ(f−1(A))

for all Borel A ⊆ Y is a probability measure on Y , called the image of µ under f . If

K is a compact second countable group with Haar measure µ and L ≤ K is a closed

subgroup, π : K → K/L the canonical surjection, then π∗µ is called Haar probability

measure on K/L. It is well-known that π∗µ is the unique K-invariant probability

measure on K/L.

2.3. Borel cocycles. Let G and H be lcsc groups, and X a standard Borel G-space

with invariant probability measure µ. A cocycle of the G-space X into H is a function

α : G×X → H such that

∀g, h ∈ G α(hg, x) = α(h, gx)α(g, x) µ-a.e.(x).

We say that α is strict iff this equation holds for all x ∈ X. If β : G × X → H is

another cocyle into H, we say that α and β are equivalent, and write α ∼ β, iff there

is a Borel map b : X → H such that

∀g ∈ G β(g, x) = b(gx)α(g, x) b(x)−1 µ-a.e.(x).

If Y is a free standard Borel H-space and f : X → Y is a Borel homomorphism

from EXG to EYH , then the map α : G ×X → H defined by α(g, x) f(x) = f(gx) is a

(strict) Borel cocycle, hereafter referred to as the cocylce corresponding to, or arising

from, f . In this case if α is equivalent to a cocycle β : G × X → H with β(g, x) =

8 SCOTT SCHNEIDER

b(gx)α(g, x) b(x)−1, then the function f ′ : X → Y defined by f ′(x) = b(x) · f(x) is a

Borel homomorphism from EXG to EYH that is a Borel reduction iff f is. We often call

f ′ an “adjustment” of f .

Throughout we shall assume that the reader is familiar with the basic machinery

of Zimmer’s superrigidity theory [22], including basic facts about Borel cocycles and

induced actions. For clear accounts of this material see Zimmer [22] or Adams-Kechris

[2].

2.4. Lattices in semisimple Lie groups. Let p be a prime. Throughout this

paper, Qp will denote the field of p-adic numbers, and Zp the ring of p-adic integers.

If G(Qp) ≤ GLn(Qp) is an algebraic Qp-group, then G(Qp) is a lcsc group with

respect to the Hausdorff topology, i.e., the topology obtained by restricting the natural

topology on Qn2

p to G(Qp). More generally, if S is a finite nonempty set of primes,

then

G =∏p∈S

G(Qp)

is a lcsc group in the product topology. Any topological notions concerning groups

of this form will always refer to this topology.

Let G be a lcsc group with left Haar measure µ, and recall that any other Haar

measure ν on G differs from µ by a scalar multiple. For each x ∈ G, the measure

µx defined by µx(E) = µ(Ex) is again a left Haar measure on G, and hence there

is ∆(x) ∈ R+ such that µx = ∆(x)µ. The function x 7→ ∆(x) is called the modular

function of G, and is easily seen to be independent of the choice of µ. It is well

known that ∆ is a continuous homomorphism from G into the multiplicative group

of positive reals. Evidently every left Haar measure on G is also right-invariant iff

∆ is identically 1; in this case G is called unimodular, and we speak simply of (bi-

invariant) Haar measure. Clearly every abelian group is unimodular; compact groups

and groups G for which G/G′ is finite are also unimodular, where here G′ is the

closure of the commutator subgroup of G. In particular, every semisimple Lie group

is unimodular.


If G is unimodular H ≤ G closed, then G/H admits a G-invariant Borel measure,

which we call a Haar measure on G/H. If G/H admits a finite invariant measure,

then H is said to have finite covolume in G, and in this case there exists a unique

invariant probability measure on G/H. A discrete subgroup Γ ≤ G of finite covolume

is called a lattice in G.

Let G be a connected semisimple Lie group with finite center, and Γ ≤ G a lattice.

Then Γ is called irreducible iff for every non-central normal subgroup N of G, Γ is

dense when projected onto G/N . In a closely related notion, an ergodic G-space

X with finite invariant measure is called irreducible iff for every non-central normal

subgroup N of G, the restricted N -action on X is still ergodic. Evidentally a lattice

Γ ≤ G is irreducible if and only if the action of G on G/Γ is irreducible.

3. A Borel Superrigidity Theorem

Given a finite, nonempty set of rational primes S = p1, . . . , ps, write OS for the

ring of integers inside the algebraic number field Q(√p1, . . . ,

√p2), and define

ΓS = SL2(OS) and ΛS = PSL2(OS).

For each A ⊆ S, let

σSA : Q(√p1, . . . ,

√ps) → R

be the field embedding that maps√pi 7→ −√pi if pi ∈ A,

√pi 7→

√pi if pi ∈ S \ A,

and is the identity on Q, so that in particular σS∅ is the inclusion embedding. Then,

identifying as usual P(S) = 2S , define

σS : ΓS →∏2S

SL2(R) by σS(γ) = 〈γ σSA〉A⊆S ,

where γ σSA has the obvious meaning and 2S is ordered, merely for definiteness, lexi-

cographically on the natural ordering for S itself.

Then if we identify ΓS with its image under σS , we have by Margulis [11, IX(1.7v)]

that

ΓS ≤ GS =∏2S

SL2(R)

10 SCOTT SCHNEIDER

is an irreducible, noncocompact, arithmetic lattice in the higher rank connected semi-

simple Lie group GS . By abuse of notation we shall also denote by σS the correspond-

ing embedding

σS : ΛS ≤ HS =∏2S

PSL2(R),

which realizes ΛS as an irreducible lattice inHS . Throughout we shall identify σS(ΓS),

σS(ΛS) with ΓS , ΛS , and rely upon context to distinguish them.

Now, given a nonempty (and possibly infinite) set of primes J , let

K(J) =∏p∈J

SL2(Zp).

Then provided√p ∈ Zq for every p ∈ S and q ∈ J , we may regard ΓS as a subgroup

of K(J) via the diagonal embedding. (This occurs, for instance, whenever S ∩ J = ∅,

2 6∈ J , and p is a quadratic residue modulo q for each p ∈ S, q ∈ J).

We then have the following Borel superrigidity theorem for the action of ΛS on full

measure subsets of that part of K(J)/Z(ΓS) on which ΛS acts freely.

Theorem 3.1. Suppose that S1, S2 are finite nonempty sets of primes and that J1,

J2 are (possibly infinite) nonempty sets of primes. For i = 1, 2, suppose that:

• √p ∈ Zq for each p ∈ Si and q ∈ Ji;

• Li ≤ K(Ji) is closed, contains Z(ΓSi), and satisfies µJi(FSi(Ji, Li)) = 1, where

µJi is Haar probability measure on K(Ji)/Li, and FSi(Ji, Li) is the subset of

K(Ji)/Li on which ΛSi acts freely;

• Xi is a µJi-measure one, ΛSi-invariant Borel subset of FSi(Ji, Li); and

• Ei is the ΛSi-orbit equivalence relation on Xi.

Suppose that f : X1 → X2 is a Borel reduction from E1 to E2. Then

(1) S1 = S2, and

(2) (K(J1)/L1,ΛS1 , µJ1) and (K(J2)/L2,ΛS2 , µJ2) are virtually isomorphic.

We now immediately prove Theorems 1.6 and 1.5 from 3.1. (These arguments are

identical to those of Thomas in [18]).


Proof of Theorem 1.6. Assuming the hypotheses of the theorem, suppose EJ1S1

≤BEJ2S2

. For i = 1, 2, let Zi = Z(ΓSi), and let Ei denote the orbit equivalence relation

arising from the free action of ΛSi on K(Ji)/Zi. Clearly E1 ≤B E2. Thus by Theo-

rem 3.1, we have that S1 = S2 and that (K(Ji)/Zi,ΛSi , µJi) are virtually isomorphic.

Arguing as in the proof of Thomas [17, 5.1], we see that K(J1)/Z1 and K(J2)/Z2 con-

tain open subgroups which are isomorphic as topological groups. By Gefter-Golodets

[7, A.6], it follows that J1 = J2.

Proof of Theorem 1.5. Assume the hypotheses of the theorem, and suppose that f :

PG(1,Qp1) → PG(1,Qp2) is a Borel reduction from Ep1S1to Ep2S2

. Recall that for i =

1, 2, Epi

Siis the orbit equivalence relation arising from the action of ΓSi on PG(1,Qpi).

Of course, since Z(ΓSi) acts trivially on PG(1,Qpi), Epi

Siis also the orbit equivalence

relation arising from the action of ΛSi on PG(1,Qpi). We now show how to apply

Theorem 3.1 to this action.

For each i = 1, 2, we regard PG(1,Qpi) as the space of 1-dimensional subspaces

of the 2-dimensional vector space Q2pi

. By Thomas [17, 6.1], SL2(Zpi) acts transi-

tively on PG(1,Qpi). Thus fixing xpi ∈ PG(1,Qpi) with (closed) stabilizer Lpi ≤

SL2(Zpi), we may identify SL2(Zpi)/Lpi as an SL2(Zpi)-set with the homogeneous

space PG(1,Qpi). Regarding ΓSi as a subgroup of SL2(Zpi), we clearly have Z(ΓSi) ≤

Lpi . Now let

PG∗(1,Qpi) = x ∈ PG(1,Qpi) | γ · x 6= x for all γ ∈ ΓSi \ Z(ΓSi)

be the subset of PG(1,Qpi) on which ΛSi acts freely. Notice that PG∗(1,Qpi) is

ΛSi-invariant. Notice also that for every x ∈ PG(1,Qpi) \ PG∗(1,Qpi), there is

γ ∈ ΓSi such that x is an eigenspace of γ. This shows that PG(1,Qpi) \ PG∗(1,Qpi)

is countable, and hence

µi(PG∗(1,Qpi)) = 1,

where µi is the Haar probability measure on PG(1,Qpi) = SL2(Zpi)/Lpi . As f is

countable-to-one, the set

x ∈ PG∗(1,Qp1) | f(x) ∈ PG(1,Qp2) \ PG∗(1,Qp2)

12 SCOTT SCHNEIDER

is countable, and so there is a µ1-conull subset X0 of PG∗(1,Qp1) such that f(X0) ⊆

PG∗(1,Qp2). As f is a Borel reduction and PG∗(1,Qp2) is ΛS2-invariant, X0 is

necessarily ΛS1-invariant.

We have now verified all the hypotheses of Theorem 3.1, and so after applying it

to the Borel reduction f , we obtain that S1 = S2 and that for i = 1, 2, the standard

Borel systems (PG(1,Qpi),ΛSi , µpi) are virtually isomorphic. By the proof of Thomas

[17, 6.3], this implies that p1 = p2.

4. Outline of Proof of Borel Superrigidity Theorem

Our proof of Theorem 3.1 will be based substantially on Thomas [18], which in

turn traces many of its main ideas to [17], [1], and [2]. The crucial ingredient in each

of these studies, as well as in our own, is superrigidity, a phenomenon in the theory of

orbit equivalence and conjugacy of ergodic group actions that has been applied quite

successfully in recent years to problems in the field of countable Borel equivalence re-

lations. Specifically, in order to prove 3.1 we shall use Zimmer’s Cocycle Superrigidity

Theorem [22, 5.2.5]. (For other important superrigidity results see, for instance, Fur-

man [6] and Popa [14], together with their applications to Borel equivalence relations

in Thomas [17] and [19], respectively). Loosely speaking, Zimmer’s theorem says that

under certain conditions on the groups G and H, the G-space X, and the cocycle

α : G ×X → H, α may be adjusted in such a way that it becomes a function of its

first variable only; i.e., that it becomes a group homomorphism α : G→ H. If Y is a

free H-space and α corresponds to a Borel homomorphism f : X → Y , then this in ef-

fect turns f into a permutation group homomorphism (f, α) : (G y X) → (H y Y ),

i.e., a Borel function f : X → Y and a group homomorphism α : G → H such that

for all g ∈ G and for almost all x ∈ X, f(g · x) = α(g) · f(x).

Without stating Zimmer’s theorem precisely at this point, we are ready to give

an extremely rough sketch of the proof of 3.1. In order to improve notation for the

proof, both here and below, we shall write S and T in place of S1 and S2, X and Y in

place of X1 and X2, and µi in place of µJi . Assuming the hypotheses of the theorem,

let f : X → Y be a Borel reduction from E1 to E2, and let α : ΛS × X → ΛT


be the cocycle corresponding to f . We would like to apply Zimmer’s theorem to

α. This would give a group homomorphism ϕ from ΛS to ΛT which would then,

hopefully, (and with a little more knowledge about ϕ), yield a contradiction due to

the incompatibility of ΛS and ΛT unless S = T .

Of course, there are a number of rather stringent hypotheses on the groups involved

that must be satisfied before we can apply Zimmer’s theorem. For this reason we

pass from ΛS y X to the induced action HS y X of the ambient connected group in

which ΛS is a lattice, with associated induced cocycle α. (Induced spaces and induced

actions will be treated in Section 5.2 below). Then we view α : HS×X → ΛT as taking

values in the larger group HT ⊇ ΛT , and finally consider the various projections

αA = αA⊆S : HS × X → PSL2(R).

We then apply Zimmer’s theorem to the cocycles αA, and obtain group homomor-

phisms ψA : HS → PSL2(R) to which the αA are equivalent. Piecing these together

will yield a single homomorphism,

ψ : HS → HT ,

to which α is equivalent when viewed as a cocycle into HT .

Our next step will be to show that ψ is injective. Eventually this will allow us to

replace ψ with one of its conjugates, ϕ, each of whose projections is nearly the identity

on some component of HS . Notice, however, that ϕ maps HS to HT , (rather than

mapping ΛS to ΛT ), and that our adjusted Borel reduction maps the HS-space X to

the HT -space Y , (rather than simply mapping X to Y , as desired). By an argument

based on Adams [1, 5.4], we will be able to come back down from the induced space

on the left side to obtain (see Figure 5):

(1) a Borel reduction f : X → Y , and

(2) the group homomorphism ϕ ΛS: ΛS → HT

such that for all γ ∈ ΛS and for µ1-a.e. x ∈ X,

f(γx) = ϕ(γ)f(x).

14 SCOTT SCHNEIDER

The argument based on Adams [1, 5.4] that allows us to pass from HS y X back

down to ΛS y X is presented in section 5.2 below.

However, we are now still faced with the problem that f takes values in Y instead

of in Y , and ϕ takes values in HT instead of in ΛT . At this point we introduce the

second key ingredient in our proof of 3.1, again following Adams [1] and Thomas [18].

Recall that Y is a product of various “twisted” copies of Y indexed by the cosets

HT /ΛT . Our strategy will be to compose our Borel reduction f with the projection

onto the second component of Y = Y ×HT /ΛT , and then to examine the image ω of

µ1 under this composite map. Faced with a similar situation in [1], Adams used a deep

measure classification result due to Ratner to show that the measure ω concentrates

on a singleton, and hence that the associated Borel reduction actually takes values

in a single twisted copy of Y lying inside Y . “Untwisting” the Borel reduction and

group homomorphism then completed the proof. Similarly, in [18] Thomas proved

a measure classification result [18, 7.3] based on the work of Margulis-Tomanov [12]

and Witte Morris [20], and used it to show in an analogous setting that the measure

ω concentrates on a finite set; untwisting in this case yielded a virtual isomorphism of

standard Borel systems. To accomplish the same purpose, in our proof we shall require

a generalization of the theorem used by Thomas in [18]. This measure classification

theorem is stated and proved below as Lemma 5.7.

As in Adams [1] and Thomas [18], our measure classification theorem actually shows

that one of two cases obtains: roughly speaking, either the measure concentrates on a

suitably small set (the desired case), or the measure is “evenly distributed” throughout

the space. In both cases it is necessary to eliminate the second possibility, which

amounts to showing that the standard Borel system (X,ΛS , µ1) does not possess

quotients of a certain form. We state this result below as Lemma 5.11.

Once we have shown that ω concentrates on a finite set Ω0 ⊆ HT /ΛT , we “untwist”

f and ϕ by an element in Ω0 so as to obtain

(1) a finite index subgroup Λ0S ≤ ΛS ;

(2) an ergodic component X ′0 ⊆ X for the action of Λ0

S on X;

(3) a group homomorphism ϕ : Λ0S → ΛT ; and


(4) a Borel map f1 : X → Y

such that for all γ ∈ Λ0S and for all x ∈ X ′

0,

ϕ(γ) · f1(x) = f1(γx).

Our next step will be to show that S ⊆ T . This will follow from the fact that

ϕ : Λ0S → ΛT embeds a conjugate of a finite index subgroup of ΛS inside ΛT . Some

matrix computations will be needed for this argument, and we prove the relevant

result below as Lemma 5.12.

Our final step will be to find the ergodic component inside Y that will be the image

of X ′0 under the desired virtual isomorphism. To achieve this it will suffice to show

that the closure of ϕ(Λ0S) has finite index in K(J2)/Z(ΓT ). In establishing this fact we

shall make use of another computational lemma, stated and proved below as Lemma

5.13. We then finish the proof of Theorem 3.1 by showing that S = T , exactly as in

Thomas [18, 8.11].

At this point some additional remarks are needed about one of the hypotheses of

Zimmer’s theorem, which we hereafter refer to as the cocycle hypothesis: that α is

not equivalent to a cocycle taking values in a proper algebraic subgroup of the target

group. In our case the target group is PSL2(R), and it is well known that every proper

algebraic subgroup of PSL2(R) is amenable. Hence this hypothesis would pose no

problem if our domain groups were Kazhdan, due to the incompatibility of actions

of Kazhdan groups with actions of amenable groups exploited in Adams-Kechris [2,

3.1]. However, ΓS is not Kazhdan, and so initially it may seem that Zimmer’s theorem

cannot be applied to the cocycles αA : HS×X → PSL2(R).1 Fortunately, while ΓS is

not Kazhdan, it does have the weaker Property (τ), and this turns out to be enough

to prove that the cocycles αA do in fact satisfy the cocycle hypothesis on Zimmer’s

theorem.2 This was the central insight of Thomas in [18]. Since ΓS (like ∆S) has

1Recall [10, 1.3] that a lattice in a locally compact group G is Kazhdan if and only if G is Kazhdan2It is clear that since ΓS has Property (τ) but is not Kazhdan ([10, Section 4.1], [23, Corollary 19],

respectively), the same is true of ΛS = ΓS/ ± 1. We use this implicitly below, as well as the fact that since

Z(ΓSi) ≤ Li in the statement of Theorem 3.1, the equivalence relation Ei = E

XiΛSi

is identical to EXiΓSi

.

16 SCOTT SCHNEIDER

Property (τ), the machinery used by Thomas in [18] can be transferred verbatim to

our setting; we shall state the necessary results below as Lemmas 5.3 and 5.4.

5. Proofs of Preliminary Lemmas

In this section we shall state and prove some of the lemmas that will be needed for

the proof of Theorem 3.1, but whose inclusion in the main body of that proof would

distract from the central line of argument.

5.1. Property (τ), E0-Ergodicity, and Cocycles into Amenable Groups. As

indicated above, in order to establish the cocycle hypothesis of Zimmer’s theorem,

it will suffice to show that each αA is not equivalent to a cocycle taking values in

an amenable subgroup of PSL2(R). We shall make essential use of the following

theorems proved by Thomas in [18] in order to establish this.

Proposition 5.1 ([18, Section 4]). If E is an E0-ergodic countable Borel equivalence

relation and F is an orbit equivalence relation arising from a Borel action of a lcsc

amenable group, then E is F -ergodic.

Proposition 5.2 ([18, 5.7]). Let K be a compact second countable group, let L ≤ K

be a closed subgoup, and let µ be the Haar probability measure on X = K/L. Let Γ

be a finitely generated dense subgroup of K. Suppose that:

(1) K is a profinite group; and

(2) Γ has Property (τ).

Then the action of Γ on (X,µ) is E0-ergodic.

As ΓS is finitely generated [11, VIII.3.3] and embeds densely (by the Strong Ap-

proximation Theorem [13, 7.12]) into the compact profinite group K(J), by the above

results we need only verify that ΓS has Property (τ). And as discussed in Lubotzky

[10, Section 4.1], ΓS does indeed have Property (τ). (This is related to the fact that

ΓS has the congruence subgroup property; see [16]).

It then follows that E1 = EXΛSis F -ergodic whenever F is an orbit equivalence

relation arising from a Borel action of a lcsc amenable group. This will enable us


to show that the cocycles αA satisfy the cocycle hypothesis of Zimmer’s theorem;

we state the precise results we shall need in the next two lemmas. The proofs are

identical to those of [18, 8.4] and [18, 8.5], respectively, and so we omit them.

Lemma 5.3. Let f : X → Y be a Borel reduction from E1 to E2, α : ΛS ×X → ΛT

the cocycle corresponding to f . Then α is not equivalent to a cocycle taking values in

an amenable subgroup of ΛT .

Lemma 5.4. For each A ⊆ T , αA is not equivalent to a cocycle taking values in a

proper algebraic R-subgroup of PSL2(R).

5.2. Induced Spaces, Actions, and Cocycles. In this section we introduce the

notions of induced space, induced action, and induced cocycle, and prove a cocycle

reduction lemma involving induced cocycles that is based on the proof of Adams

[1, 5.4] and will play a key role in the proof of Theorem 3.1. For clear accounts of

cocycles and induced spaces see [2, Section 2] and [1, Section 5]. We remark that all

our cocycles are Borel.

Let G be a lcsc group and Γ a lattice in G, so that G/Γ admits a G-invariant Borel

probability measure, ν. Fix a Borel transversal T ⊆ G for G/Γ containing 1G, and

identify T with G/Γ via the natural identification t↔ tΓ, so that we may view T as

a G-space with invariant probability measure ν and natural G-action defined by

g.t = the unique element of T in the coset gtΓ.

Also define the associated cocycle ρ : G× T → Γ by

ρ(g, t) = the unique γ ∈ Γ such that (g.t)γ = gt.

Further suppose that X is a standard Borel Γ-space with invariant Borel probability

measure µ. We then define the induced space (X, µ) to be

(X, µ) = (X ×G/Γ, µ× ν),

and the induced action of G on (X, µ) by

g ∗ (x, t) = (ρ(g, t) · x, g.t).

18 SCOTT SCHNEIDER

It is easily checked (see [2]) that µ = µ × ν is a G-invariant probability measure on

X that is ergodic if µ is.

Now let H be any group and suppose that α : Γ×X → H is a strict cocycle into

H. Then we call the map

α : G× X → H

defined by

α(g, 〈x, t〉) = α(ρ(g, t), x)

the cocycle induced from α (or simply the induced cocycle).

We shall now prove the following cocycle reduction result involving induced cocy-

cles. Our proof is based on that of Adams [1, 5.4], and corrects, at Adams’ suggestion,

the slight error found there. In our proof we sometimes suppress reference to t when

denoting elements of X, writing x in place of 〈x, t〉 when no confusion can arise.

Lemma 5.5. Let Γ ≤ G, T , (X,µ), and (X, µ) be as above, suppose α : Γ×X → H

is a cocycle into an arbitrary group H, and let α : G× X → H be the cocycle induced

from α. Suppose there is a group homomorphism ϕ : G→ H such that α is equivalent

to a cocycle β : G× X → H satisfying

β(g, x) = ϕ(g) for all g ∈ G and for µ-a.e. x ∈ X.

Then α is equivalent to a cocycle β : Γ×X → H satisfying

β(γ, x) = ϕ(γ) for all γ ∈ Γ and for µ-a.e. x ∈ X.

Proof. We begin with the precise statement of the equivalence α ∼ β. Thus let

b : X → H be a Borel function such that

(∀g ∈ G)(∀∗x ∈ X) ϕ(g) = b(g ∗ x) α(g, x) b(x)−1.

By Fubini-Tonelli, this implies

(∀g ∈ G)(∀∗x ∈ X)(∀∗t ∈ T ) ϕ(g) = b(g ∗ 〈x, t〉

)α(g, 〈x, t〉

)b(〈x, t〉

)−1.


Replacing g with tgt−1 and then restricting the first quantifier to range over Γ ≤ G,

we obtain

(∀γ ∈ Γ)(∀∗x ∈ X)(∀∗t ∈ T ) ϕ(tγt−1) = b(tγt−1 ∗ 〈x, t〉

)α(tγt−1, 〈x, t〉

)b(〈x, t〉

)−1.

Again applying Fubini-Tonelli (and using the fact that Γ is countable), we may write

(∀∗t ∈ T )(∀γ ∈ Γ)(∀∗x ∈ X) ϕ(tγt−1) = b(tγt−1 ∗ 〈x, t〉

)α(tγt−1, 〈x, t〉

)b(〈x, t〉

)−1.

Now fix such a t ∈ T for which the above holds. Then simplifying we have

(∀γ ∈ Γ)(∀∗x ∈ X) ϕ(tγt−1) = b(〈γx, t〉

)α(γ, x) b

(〈x, t〉

)−1,

and hence

(∀γ ∈ Γ)(∀∗x ∈ X) ϕ(γ) = ϕ(t)−1b(〈γx, t〉

)α(γ, x) b(〈x, t〉)−1ϕ(t).

Now if we define the Borel function b : X → H by

b(x) = ϕ(t)−1b(〈x, t〉)

for all x ∈ X, then we have for all γ ∈ Γ and for µ-a.e. x ∈ X,

ϕ(γ) = b(γx)α(γ, x) b(x)−1,

as desired.

5.3. A Measure Classification Theorem. In our proof of Theorem 3.1 we shall

need to appeal at one point to a measure classification result in order to show that our

adjusted Borel reduction into the induced space Y takes values in only finitely many

of the copies of Y lying inside Y . This technique goes back to Adams, who made

use in [1] of Ratner’s measure classification theorem [15]. Our approach is based on

Thomas [18], who used a measure classification result essentially due to David Witte

Morris [20, 5.8]. We recall first the definition of an algebraic probability measure.

Definition 5.6. Let H be a lcsc group and let L be a closed subgroup of H. Then a

probability measure µ on H/L is said to be algebraic iff there exists a closed subgroup

C of H such that

(1) µ is C-invariant; and

20 SCOTT SCHNEIDER

(2) µ is supported on a C-orbit; i.e., there exists x ∈ H/L such that µ(Cx) = 1.

Lemma 5.7. Let ϕ : HS → HT be an injective group homomorphism. Then every

ϕ(ΛS)-invariant, ϕ(ΛS)-ergodic probability measure on HT /ΛT is algebraic.

Proof. We shall need two results, the first of which is Theorem 3.1 in Margulis-

Tomanov [12]:

Proposition 5.8. Let G be an almost linear group, Γ ≤ G a closed subgroup, H ≤ G

a subgroup generated by its unipotent algebraic subgroups, and µ an H-invariant,

ergodic probability measure on G/Γ. Then µ is algebraic.

Now, write H = ϕ(HS) and let ∆ : H → H × HT be the diagonal embedding.

Note that ∆(H) ∼= H ∼= HS is generated by its unipotent algebraic subgroups inside

H ×HT . Hence by 5.8, every ∆(H)-invariant, ergodic probability measure on

(H ×HT )/(ϕ(ΛS)× ΛT ) = (H/ϕ(ΛS))× (HT /ΛT )

is algebraic — and this is precisely the third hypothesis needed in the following result

of Witte Morris (see [18, A.1]):

Proposition 5.9 (Witte Morris). Let G be a lcsc group and let H, L be closed

subgroups of G. Suppose that:

(1) Γ is a lattice in H;

(2) ∆ : H → H ×G is the diagonal embedding; and

(3) every ergodic ∆(H)-invariant probability measure on (H/Γ) × (G/L) is alge-

braic.

Then every ergodic Γ-invariant probability measure on G/L is algebraic.

It follows immediately that every ϕ(ΛS)-invariant, ergodic probability measure on

HT /ΛT is algebraic, completing the proof of the Lemma 5.7.

5.4. Dynamics of ΓS Actions. In order to show that ω is finitely supported, we

shall in particular need to eliminate the possibility that ω is “evenly distributed”

throughout HT /ΛT by showing that (X,ΛS , µ1) does not admit quotients of a certain

form (cf [18, 6.4]). Thus we shall need the notion of a quotient of a Borel system.


Definition 5.10. For each i = 1, 2, let Γi be a countable group and let Xi be

a standard Borel Γi-space with an invariant ergodic probability measure µi. Then

(X2,Γ2, µ2) is a quotient of (X1,Γ1, µ1) iff there exist

(1) a surjective homomorphism ϕ : Γ1 → Γ2 and

(2) a Borel function f : X1 → X2

such that the following conditions are satisfied:

(1) f∗µ1 = µ2; and

(2) for each γ ∈ Γ1, f(γ · x) = ϕ(γ) · f(x) for µ1-a.e. x ∈ X1.

If ϕ is an isomorphism then we call (X2,Γ2, µ2) a factor of (X1,Γ1, µ1); if in addition

f is injective and has µ2-conull range, we call (Xi,Γi, µi) isomorphic.

We are now ready to record our desired result.

Lemma 5.11. Suppose that H is isomorphic to a finite product of copies of the group

PSL2(R). Let M be a proper closed subgroup of H such that H/M has finite volume,

and let m be Haar probability measure on H/M . Let ∆ be an irreducible lattice in H.

Then (H/M,∆,m) is not a quotient of (K(J1)/L1,ΛS , µ1).

Proof. By Adams [1, 6.3] together with Zimmer [22, 2.2.20], the action of H on H/∆

is strongly mixing. It follows that we may proceed exactly as in Thomas [18, 6.4]. In

fact, since ∆ is already an irreducible lattice in H, our argument is properly contained

in that of [18, 6.4], and we direct the reader to this reference for the details.

5.5. Some Computations Involving Lattices in PSL2(R). We shall use the fol-

lowing two lemmas towards the end of our proof of Theorem 3.1 in order to detect

the primes that distinguish ΛS from ΛT , and hence E1 from E2.

Lemma 5.12. Suppose v =

a b

c d

∈ PSL2(R) conjugates a finite index subgroup

of ΛS into ΛT .3 Then there exists a positive integer k such that ka2, kb2, kc2, kd2,

kab, kac, kad, kbc, kbd, kcd ∈ OT .

3Recall that ΛS = ΓS/Z(ΓS), where Z(ΓS) = ±1. Thus, both here and below, we shall denote elements

of ΛS (and ΛT ) by 2× 2 matrices (aij) which we remember to identify with (−aij).

22 SCOTT SCHNEIDER

Proof. Let Λ0S be a finite index subgroup of ΛS such that vΛ0

Sv−1 ≤ ΛT . Since

[ΛS : Λ0S ] <∞, there exist positive integers k0, k1, and k2 such that 1 k1

0 1

, 1 0

k2 1

, 1− k2 −k2

k2 1 + k2

∈ Λ0S .

Let k′ = k0k1k2, so that 1 k′

0 1

, 1 0

k′ 1

, 1− k′ −k′

k′ 1 + k′

∈ Λ0S .

Then from a b

c d

1 k′

0 1

d −b

−c a

=

1− k′ac k′a2

−k′c2 1 + k′ac

∈ ΛT

and a b

c d

1 0

k′ 1

d −b

−c a

=

1 + k′bd −k′b2

k′d2 1− k′bd

∈ ΛT ,

we get

k′a2, k′b2, k′c2, k′d2, k′ac, k′bd ∈ OT .

Next consider a b

c d

1− k′ −k′

k′ 1 + k′

d −b

−c a

=

(1− k′)ad+ k′ac+ k′bd− (1 + k′)bc −(1− k′)ab− k′a2 − k′b2 + (1 + k′)ab

(1− k′)cd+ k′c2 + k′d2 − (1 + k′)cd −(1− k′)bc− k′ac− k′bd+ (1 + k′)ad

.From the upper left, and remembering that k′ac, k′bd ∈ OT , we obtain

−k′(ad+ bc) ∈ OT .

Of course, since ad− bc = 1, we also have

−k′(ad− bc) ∈ OT ,

and then combining these equations gives

2k′ad, 2k′bc ∈ OT .


From the lower left, and remembering that k′c2, k′d2 ∈ OT , we get

2k′cd ∈ OT ,

and similarly, from the upper right,

2k′ab ∈ OT .

Thus letting k = 2k′ completes the proof.

Lemma 5.13. Assume the hypotheses of Lemma 5.12. Let kT denote the field of

fractions of OT . Then v ∈ PSL2(kT ).

Proof. By 5.12, v normalizes PSL2(kT ), so it will suffice to show that PSL2(kT ) is

its own normalizer in PSL2(R). Notice first that every element s in the normalizer

of PSL2(kT ) will induce, via conjugation, an automorphism τs of PSL2(kT ). Since

conjugation cannot induce automorphisms of PSL2(kT ) arising from field automor-

phisms, and the only other outer automorphism of PSL2(kT ), namely multiplication

by 0 −1

1 0

,also does not come from conjugation, each τs must in fact be an inner automorphism

induced by an element of PSL2(kT ) itself. But if two elements of the normalizer

of PSL2(kT ), one in PSL2(kT ) itself, give rise to the same conjugation, then their

difference centralizes PSL2(kT ). Since only the identity centralizes PSL2(kT ), both

elements must have come from PSL2(kT ), and hence PSL2(kT ) is its own normalizer.

This shows that v ∈ PSL2(kT ).

6. Proof of Borel Superrigidity Theorem

We now commence with the proof of Theorem 3.1. To improve notation, throughout

we write S and T in place of S1 and S2, X and Y in place of X1 and X2, and µi in

place of µJi . Assuming the hypotheses of the theorem, let

α : ΛS ×X → ΛT

24 SCOTT SCHNEIDER

be the (strict) Borel cocycle defined by

α(γ, x) · f(x) = f(γx)

for all γ ∈ ΛS , x ∈ X, so that α is the cocycle corresponding to the Borel reduction

f : X → Y into the free ΛT -space Y , as illustrated by Figure 1.

ΛSx

γ

yγx

(X,µ1)

f−−−−→

ΛTf(x)

α(γ,x)

yf(γx)

(Y, µ2)

Figure 1

Next consider the induced action of HS on X = X ×HS/ΛS , and let

α : HS × X → ΛT

be the corresponding cocycle induced from α. Let ν be the Haar probability measure

on HS/ΛS and let µ1 = µ1×ν. Then α corresponds to the Borel reduction f : X → Y

defined by f(〈x, t〉) = f(x), as illustrated in Figure 2.

HS

x

g

yg ∗ x

(X, µ1)

f−−−−→

ΛTf(x)

α(g,x)

yf(g ∗ x)

(Y, µ2)

Figure 2

Now, let σT : ΛT → HT be the embedding that realizes ΛT as an irreducible lattice

in HT , and for each A ⊆ T , let πTA : HT → PSL2(R) be the canonical projection onto

the factor of HT corresponding to A. Define the cocycles

αA = πTA σT α : HS × X → PSL2(R).


By Lemma 5.4, any cocycle equivalent to some αA is Zariski dense in PSL2(R).

Furthermore, by Zimmer [21, 2.4], X is an irreducible ergodic HS-space. The re-

maining hypotheses of Zimmer’s superrigidity theorem [22, 5.2.5] are easily verified,

and hence we may apply it to the cocycles αA to obtain, for each A ⊆ T , a rational

R-homomorphism

ψA : HS → PSL2(R)

such that αA is equivalent to the cocycle

βA : HS × X → PSL2(R)

defined by

βA(g, x) = ψA(g) for all g ∈ HT , x ∈ X.

For each A ⊆ T , let the Borel map hA : X → PSL2(R) witness the equivalence of

αA with βA, so that we have, for all g ∈ HS and for µ1-a.e. x ∈ X,

ψA(g) = βA(g, x) = hA(g ∗ x) · αA(g, x) · hA(x)−1.

Define

ψ : HS → HT

by

ψ(g) = 〈ψA(g)〉A⊆T ,

and define

h : X → HT

by

h(x) = 〈hA(x)〉A⊆T .

Then we have, for all g ∈ HS and for µ1-a.e. x ∈ X,

ψ(g) = β(g, x) := h(g ∗ x) · (σT α)(g, x) · h(x)−1,

so that σT α is equivalent to β with witness h : X → HT , as illustrated in Figure 3.

Our next goal will be to show that ψ : HS → HT is injective. This will allow us to

replace ψ with a simpler conjugate of ψ, and will furthermore imply that the image of

26 SCOTT SCHNEIDER

HS

x

g

yg ∗ x

(X, µ1)

fˆ→

ef = h ∗fˆ→

HT

f (x)h(x)−−−−→ f(x)

(σT α)(g,x)

y yψ(g)=β(g,x)

f (g ∗ x) h(g∗x)−−−−→ f(g ∗ x)

(Y , µ2)

Figure 3

ΛS under this conjugate remains an irreducible lattice in ϕ(HS). Towards this end,

define the Borel function fˆ: X → Y by

f (〈x, t〉) = 〈f(x), 1〉.

It is easily verified that fˆ is a Borel reduction from EXHSto EYHT

and that fˆcorre-

sponds to the cocycle σT α. Further define f : X → Y to be the adjusted Borel

reduction f = h ∗ f , so that f corresponds to the cocycle β(g, x) = ψ(g). Then we

have, for all g ∈ HS and for µ1-a.e. x ∈ X,

ψ(g) ∗ f(x) = f(g ∗ x).

Figure 3 may be helpful in understanding the proof of the following Lemma.

Lemma 6.1. ψ : HS → HT is injective and |S| ≤ |T |.

Proof. Recall that HS =∏A⊆S PSL2(R) and HT =

∏A⊆T PSL2(R). In what follows

we shall need some notation to refer to the factors in HS , HT . Previously we defined,

for A ⊆ T , πTA : HT → PSL2(R) to be the canonical projection of HT onto the factor

corresponding to A ⊆ T . We define πSA : HS → PSL2(R) similarly, and remark that

we shall sometimes write PSL2(R)A⊆S in place of πSA(HS), and PSL2(R)A⊆T in place

of πTA(HT ).

Now, notice that each ψA maps HS , a finite product of (at least two) copies of

PSL2(R), into PSL2(R). Since PSL2(R) is simple, the only normal subgroups of

HS = PSL2(R)× · · · × PSL2(R)


are products of its factors, ie, subgroups of the form

N∅ × · · · ×NS ,

where each NA⊆S is either PSL2(R) or the trivial group. By Lemma 5.4, ψA is

nontrivial; therefore, since ψA is continuous and hence must preserve dimension, its

kernel must be all but one of the factors in HS . In other words,

ψA : PSL2(R)× · · · × PSL2(R) → PSL2(R)

is really only a function of one of the factors in HS . Let A′ ⊆ S correspond to

the factor in HS on which ψA is nontrivial, so that ψA is really only a function

of PSL2(R)A′⊆S . We claim that the restriction of ψA to this factor is an isomor-

phism from PSL2(R)A′⊆S to PSL2(R)A⊆T . Injectivity follows from the simplicity of

PSL2(R), and surjectivity from the fact that the continuous map

ψA PSL2(R)A′⊆S

preserves dimension, where a full dimension subgroup of PSL2(R) would have finite

index and PSL2(R) has no subgroups of finite index. Thus to each ψA, (A ⊆ T ), we

can associate a subset A′ ⊆ S corresponding to the single factor in HS on which ψA

is an isomorphism. We now show that for each factor in HS there is some ψA which

is non-trivial on that factor; ie, that the association A 7→ A′ is a surjection of P(T )

onto P(S).

Suppose not; that is, suppose there is some factor PSL2(R) of HS that lies in the

kernel of each ψA, A ⊆ T . We view this factor, call it N , as a normal subgroup

of HS , and consider its restricted action on X, remembering that ψ(g) = 1 for all

g ∈ N . Recall that by Zimmer [21, 2.4], X is an irreducible HS-space, which means

that N acts ergodically on X. However, since ψ(g) = 1 for all g ∈ N , we have that

the adjusted Borel function f : X → Y is a Borel reduction from EXN to EYHTsuch

that for µ1-a.e. x ∈ X,

xEXN y iff f(x) = f(y).

Clearly this contradicts the ergodicity of N on X. Thus each factor PSL2(R) in HS

is realized as PSL2(R)A′⊆S for some A ⊆ T , and for any A ⊆ T the restriction of

28 SCOTT SCHNEIDER

ψA to PSL2(R)A′⊆S is an isomorphism. It follows easily that ψ is injective and that

|S| ≤ |T |.

In the above proof we saw that each ψA can be viewed as an isomorphism of

PSL2(R)A′ ≤ HS onto PSL2(R)A ≤ HT . Specifically, fix A ⊆ T , let

x = 〈x∅, . . . , xS〉 ∈ HS =∏2S

PSL2(R),

and let xA′ = πSA′(x), where

ψA PSL2(R)A′⊆S: PSL2(R)A′⊆S → PSL2(R)A⊆T

is an isomorphism. As every automorphism of PSL2(R) is conjugation by some

element of PGL2(R), there is gA ∈ PGL2(R) such that for all x ∈ HS ,

ψA(x) = gAxA′g−1A .

We now conjugate away as much of gA as possible. Specifically, we define a new

injective rational R-homomorphism ϕA : HS → PSL2(R)A⊆T , a conjugate of ψA, as

follows:

If r = det gA > 0, define ϕA : HS → PSL2(R)A⊆T by

ϕA(x) =

r1/2 0

0 r1/2

g−1A ψA(x)gA

r−1/2 0

0 r−1/2

.In this case notice that ϕA is a conjugate of ψA by an element of PSL2(R), and that

ϕA(x) = xA′ for all x ∈ HS .

If, on the other hand, r = det gA < 0, define ϕA : HS → PSL2(R)A⊆T by

ϕA(x) =

−(−r)1/2 0

0 (−r)1/2

g−1A ψA(x)gA

(−r)−1/2 0

0 −(−r)−1/2

.In this case ϕA is again a conjugate of ψA by an element of PSL2(R), and if

xA′ =

a b

c d

for x ∈ HS ,


then we have

ϕA(x) =

−a b

c −d

.Write

gA =

r1/2 0

0 r1/2

g−1A or

−(−r)1/2 0

0 (−r)1/2

g−1A ,

depending, as above, on whether det gA is positive or negative. Then

ϕA = gAψAg−1A ,

with gA ∈ PSL2(R). Supposing we have defined gA in this manner for each A ⊆ T ,

we let

g = 〈gA〉A⊆T ∈ HT ,

and define

ϕ = gψg−1 : HS → HT .

Then ϕ is an injective homomorphism from HS to HT such that for all γ ∈ ΛS ,

each component of ϕ(γ) is either γσSA , or γσ

SA with main diagonal scaled by −1, for

some Galois automorphism σSA.

Of course, we must now make the necessary adjustment to f , as well. Define

b : X → HT

by

b(x) = gh(x),

so that (σT α) is equivalent, via b, to the cocycle β1 : HS × X → HT defined by

β1(g, x) = ϕ(g).

Thus we have for all g ∈ HS and for µ1-a.e. x ∈ X,

ϕ(g) = β1(g, x) = b(g ∗ x) · (σT α)(g, x) · b(x)−1.

Then define

f1 : X → Y

30 SCOTT SCHNEIDER

to be the adjusted Borel reduction

f1 = b ∗ f ,

so that f1 corresponds to β1, giving us the picture illustrated in Figure 4 below.

HS

x

g

yg ∗ x

(X, µ1)

fˆ→

ef1 = b ∗fˆ→

HT

f (x)b(x)−−−−→ f1(x)

(σT α)(g,x)

y yϕ(g)= β1(g,x)

f (g ∗ x) b(g∗x)−−−−→ f1(g ∗ x)

(Y , µ2)

Figure 4

Now, following Adams [1, 5.4], we wish to come back down on the left side from the

induced HS-space X to the original ΛS-space X. Indeed, by Lemma 5.5 the cocycle

σT α : ΛS ×X → HT is equivalent to the cocycle

β : ΛS ×X → HT

defined by

β(γ, x) = ϕ(γ).

We now use the cocycle equivalence σT α ∼ β to obtain an adjusted Borel

reduction from X to Y that corresponds to the group homomorphism

ϕΛS: ΛS → HT .

First define j : Y → Y by j(y) = 〈y, 1〉, so that j is ΛT -equivariant, i.e., j(γ · y) =

γ ∗ j(y) for all γ ∈ ΛT , y ∈ Y . Then define

fˆ= j f : X → Y ,

so that fˆcorresponds to σT α. Further let b : X → HT witness the equivalence of

σT α with β, as in the proof of 5.5, so that for all γ ∈ ΛS and µ1-a.e. x ∈ X,

β(γ, x) = b(γx)(σT α)(γ, x)b(x).


Finally, define

f = b ∗ f ,

so that f corresponds to β and hence for all γ ∈ ΛS and for µ1-a.e. x ∈ X, we have

(5.2) f(γx) = β(γ, x) = ϕ(γ)f(x).

Let us pause now to summarize our progress. From the original Borel reduction

f : X → Y from E1 to E2 with corresponding cocycle α : ΛS × X → ΛT , we have

obtained a new Borel reduction f : X → Y corresponding to the cocycle β(γ, x) :

ΛS ×X → HT defined by β(γ, x) = ϕ(γ). This is illustrated in Figure 5.

ΛSx

γ

yγx

(X,µ1)

f−−−−→

HT

f(x)

ϕ(γ)

yf(γx)

(Y , µ2)

Figure 5

Our only problem now is that f takes values in Y instead of in Y (and ϕ takes

values in HT instead of in ΛT ). Recall that Y is a product of various twisted copies

of Y indexed by the cosets of ΛT in HT . Our next goal will be to show that, µ1-a.e.,

f takes values in only finitely many of the copies of Y lying inside Y . As in Adams

[1] and Thomas [18], we shall accomplish this by projecting f(X) onto the second

coordinate of Y and then applying 5.7 to the resulting image of µ1, making essential

use of 5.11 in the process.

Thus let η : Y → HT /ΛT be the projection onto the second coordinate, and let

ω = (η f)∗µ1.

Then ω is a ϕ(ΛS)-invariant, ϕ(ΛS)-ergodic probability measure on the homogeneous

HT -space HT /ΛT . We proceed now to show that ω is finitely supported. Define

H = ϕ(HS) ≤ HT ,

32 SCOTT SCHNEIDER

and note that ϕ(ΛS) is an irreducible lattice in H.

Lemma 6.3. ω is supported on a finite set Ω0 ⊆ HT /ΛT .

Proof. The proof is identical to that of Thomas [18, 8.7], using Lemma 5.7 in place

of [18, 7.3] and 5.11 in place of [18, 6.4].

Since Ω0 is ϕ(ΛS)-invariant and finite, there exists a finite index subgroup of ΛS ,

call it Λ0S , whose image under ϕ acts trivially on Ω0.

In order to understand the next stage in our proof, it may be helpful to refer to

Figure 6. Intuitively, we would like to untwist ϕ so that it sends Λ0S into ΛT . Thus

Λ0S ≤f.i.

ΛS

x

γ

yγx

(X,µ1)

(f , ϕ)→

(f ϕ)→

ϕ(Λ0S) ≤ ΛT ≤ HT

f(x) u−1

−−−−→ f(x)

ϕ(γ)

y yϕ(γ)

f(γx) u−−−−→ f(γx)

(Y = Y ×HT /ΛT , µ2)

η−−−−→

ϕ(Λ0S) ≤

f.i.ϕ(ΛS)

uΛT ∈ Ω0

(Ω0, ω) ⊆ HT /ΛT

Figure 6

we fix uΛT ∈ Ω0, (recalling that u ∈ U ⊆ HT ), and then define f : X → Y by

f(x) = u−1 ∗ f(x),

and ϕ : HS → HT by

ϕ(g) = u−1ϕ(g)u.

Then ϕ(Λ0S) fixes ΛT ∈ HT /ΛT , so we must have ϕ(γ) ∈ ΛT for all γ ∈ Λ0

S . But

recall that here we are treating ΛT ≤ HT and Λ0S ≤ ΛS ≤ HS as σT - and σS-

diagonal subgroups, respectively, and hence this implies that there exists a fixed

homomorphism ϕ : Λ0S → ΛT such that for all A ⊆ T and for all γ ∈ Λ0

S ,

ϕ(γ)σTA = πA(u−1)(ϕA σS)(γ)πA(u).

To be absolutely precise, we define, for γ ∈ Λ0S ,

ϕ(γ) = (π∅ ϕ σS)(γ).


Further define

v = u−1∅ ∈ PSL2(R),

so that for all γ ∈ Λ0S , we have ϕ(γ) = vϕ∅(γ)v−1, where, as noted before, ϕ∅(γ)

is either γσSA , or γσ

SA with main diagonal scaled by −1, σSA a Galois automorphism.

As ϕ(Λ0S) is clearly a subgroup of finite index in ΛS , this implies that v ∈ PSL2(R)

conjugates a finite index subgroup of ΛS into ΛT .

Now let f1 : X → Y be the Borel map obtained by projecting f(x) ∈ Y =

Y × (HT /ΛT ) into Y , and define

X0 = x ∈ X | (η f)(x) = uΛT .

Then µ1(X0) = ω(uΛT ) = 1/|Ω0| > 0. Moreover X0 is Λ0S-invariant, and for all

γ ∈ Λ0S and µ1-a.e. x ∈ X0,

(5.6) ϕ(γ) · f1(x) = f1(γx).

Now, since µ1(X0) > 0 and the only µ1-positive measure, Λ0S-invariant subsets of

X are unions of ergodic components for the action of Λ0S on X, there exists an ergodic

component X ′0 ⊆ X0 for the action of Λ0

S on X. Recall that we are attempting to

show that the standard Borel systems

(5.7) (X,ΛS , µ1) ∼= v.i. (Y,ΛT , µ2)

are virtually isomorphic. It is worth noting that we have now found the positive

measure subsetX ′0 ⊆ X, finite index subgroup Λ0

S ≤ ΛS , and injective homomorphism

ϕ that will end up witnessing this virtual isomorphism (5.7).

There now remain two crucial steps in our proof. We wish to show that S ⊆ T , and

that the closure of ϕ(Λ0S) has finite index in K(J2)/Z(ΓT ). The latter will enable us

to find the ϕ(Λ0S)-invariant, positive µ2-measure subset of Y that will end up being

the image of X ′0 under the virtual isomorphism (5.7). Establishing the former will be

a crucial step in obtaining (5.7), and in eventually showing S = T .

Proposition 6.5. S ⊆ T .

34 SCOTT SCHNEIDER

Proof. Let p ∈ S. Notice that m+ n√p 0

0 m− n√p

∈ ΛS ⇐⇒ m2 − n2p = 1.

By the theory of the Pell Equation (for instance, see [3]), there exist infinitely many

solutions to the diophantine equation x2 − py2 = 1, and hence there are infinitely

many elements of the form m+ n√p 0

0 m− n√p

in ΛS . As ϕ(Λ0

S) has finite index in ΛS , this gives us infinitely many elements of the

above form in ϕ(Λ0S); thus fix integers m, n such that m+ n

√p 0

0 m− n√p

∈ ϕ(Λ0S).

But now if we write v =

a b

c d

, then we have

a b

c d

m+ n√p 0

0 m− n√p

d −b

−c a

=

m+ (ad+ bc)n√p −2abn

√p

2cdn√p m− (ad+ bc)n

√p

∈ ΛT .

The result now follows immediately from Lemma 5.12.

Proposition 6.6. The closure of ϕ(Λ0S) has finite index in K(J2)/Z(ΓT ).

Proof. First note that since PSL2(Zp) is open in PSL2(Qp) and g ·PSL2(Q) is dense

in PSL2(Qp) for each g ∈ PSL2(Qp), any element of PSL2(Qp) can be written as

a product of an element of PSL2(Zp) with an element of PSL2(Q). Thus for each

pi ∈ J2, write v = aibi, with ai ∈ PSL2(Zpi) and bi ∈ PSL2(Q). Now write v


for the image of v under the diagonal embedding into∏p∈J2

PSL2(Zp), and write

a = 〈ai〉pi∈J2 , b = 〈bi〉pi∈J2 , so that v = ab. Finally, write

Φ =(b · ϕ(Λ0

S) · b−1) ⋂ ∏

p∈J2

PSL2(Z),

and Φ for the closure of Φ in PSL2(Qp).

Now, we claim that the entries of v are actually in Zp for all but finitely many

pi ∈ J2, and thus we may take bi = 1 for all but finitely many i. To see this, let α ∈ kTbe an arbitrary entry in the matrix of v, and notice (by 5.13) that after rationalizing

the denominator we may write α = α′/d, with α′ ∈ Z[√p1, . . . ,

√pt] ⊆ OT and d ∈ N.

Now, for each p ∈ J2 we have Z[√p1, . . . ,

√pt] ⊆ Zp by hypothesis; it is easy to check,

then, that α = α′/d ∈ Zp iff p - d. As only finitely many primes can divide d, the

claim follows.

Since PSL2(Q) commensurates PSL2(Z), this implies that ∏p∈J2

PSL2(Z) : Φ

<∞,

and hence, passing to the closures, that ∏p∈J2

PSL2(Zp) : Φ

<∞.

But then further conjugating by a ∈∏p∈J2

PSL2(Zp) leaves∏p∈J2

PSL2(Zp) fixed

and preserves the finite index of Φ; hence

aΦa−1 = aΦa−1 = ϕ(Λ0S) ≤f.i.

∏p∈J2

PSL2(Zp),

as desired.

This shows that ϕ(Λ0S) has finitely many orbits on K = K(J2)/Z(ΓT ), and hence

finitely many orbits on the homogeneous K-space K(J2)/L2, call them Z1, . . . , Zn.

Then each Zi is compact second-countable, carries unique invariant (Haar) probabil-

ity measure, and is isomorphic as a K-space to K/(Stab zi), zi ∈ Zi fixed. Hence, by

[18, 2.2(a)], the action of ϕ(Λ0S) on each of these orbits will be uniquely ergodic. Fur-

thermore, as each of these orbits is ϕ(Λ0S)-invariant, the Borel map h : X ′

0 → Zii<n

36 SCOTT SCHNEIDER

sending x ∈ X ′0 to the orbit Zi containing f1(x) is Λ0

S-invariant; hence by the ergod-

icity of Λ0S on X ′

0 it is (µ1)X′0-a.e. constant, and so by passing to a (µ1)X′

0-measure

one subset of X ′0 if necessary, we may assume that there exists an orbit Zi0 such

that f1(X ′0) ⊆ Zi0 . Write Y ′0 = Zi0 ∩ Y . Since Λ0

S preserves the probability measure

(µ1)X′0

on X ′0, it follows that ϕ(Λ0

S) preserves the probability measure (f1)∗(µ1)X′0

on

Y ′0 , as the latter is just the image of (µ1)X′0

through a homomorphism of permutation

groups, (f1, ϕ) : (X ′0,Λ

0S) → (Y ′0 , ϕ(Λ0

S)). Since the action of ϕ(Λ0S) on Y ′0 is uniquely

ergodic, this implies that (f1)∗(µ1)X′0

= (µ2)Y ′0 . Hence, after deleting a µ2-null sub-

set of Y ′0 if necessary, we may assume that f1(X ′0) = Y ′0 . It is now evident that the

standard Borel systems (X ′0,Λ

0S , (µ1)X′

0) and (Y ′0 , ϕ(Λ0

S), (µ2)Y ′0 ) are isomorphic.

The proof of Theorem 3.1 may now be completed by showing S = T exactly as in

Thomas [18, 8.9].

References

[1] S. Adams, Containment does not imply Borel reducibility, in: Set Theory: The Jahnal Conference (Ed:

S. Thomas), DIMACS Series, vol. 58, American Mathematical Society, 2002, pp. 1-23.

[2] S. Adams and A. S. Keckris, Linear algebraic groups and countable Borel equivalence relations, J. Amer.

Math. Soc. 13 (2000), 909-943.

[3] E. Barbeau, Pell’s Equation, Springer, 2003.

[4] M. B. Bekka and M. Mayer, Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous

Spaces, London Math. Soc. Lecture Notes Series, Vol. 269 (Cambridge University Press, 2000).

[5] J. Feldman and C. C. Moore, Ergodic equivalence relations and Von Neumann algebras I, Trans. Amer.

Math. Soc. 234 (1977), 289-324.

[6] A. Furman, Orbit equivalence rigidity, Ann. of Math., 150 (1999), 1083-1108.

[7] S. L. Gefter and V. Ya. Golodets, Fundamental groups for ergodic actions and actions with unit funda-

mental groups, Publ. RIMS, Kyoto Univ. 24 (1988), 821-847.

[8] S. Jackson, A. S. Kechris, and A. Louveau, Countable Borel equivalence relations, J. Math. Logic, 2

(2002), 1-80.

[9] A. S. Kechris, Countable sections for locally compact group actions, Ergodic Theory Dynamical Systems,

12 (1992), 283-295.

[10] A. Lubotzky, On Property (τ), in preparation, available on web.

[11] G. A. Margulis, Discrete Subgroups of Semisimple Lie Groups, Erg. der Math. und ihrer Grenz. 17,

Springer-Verlag, 1991.

[12] G. A. Margulis and G. M. Tomanov, Measure rigidity for almost linear groups and its applications,

Journal d’Analyse Math., 69 (1996), 25-54.


[13] V. Platonov and A. Rapinchuk, Algebraic Groups and Number Theory, Academic Press, 1994.

[14] S. Popa, Cocycle and orbit equivalence superrigidity for malleable actions of ω-rigid groups, Inventiones

Mathematicae, 170 (2) (2007), 243-295.

[15] M. Ratner, On Raghunathan’s measure conjecture, Ann. of Math. 134 (1991), 545-607.

[16] J.P. Serre, Le problem des groupes de congruence pour SL2, Ann. Math. 92 (1970), 489-527.

[17] S. Thomas, Superrigidity and countable Borel equivalence relations, Annals Pure Appl. Logic, 120 (2003),

237-262.

[18] S. Thomas, Property (τ) and countable Borel equivalence relations, J. Math. Logic 7 (2007), 1-34.

[19] S. Thomas, Popa superrigidity and countable Borel equivalence relations, accepted for publication in Ann.

Pure Appl. Logic as of 7/31/07.

[20] D. Witte, Measurable quotients of unipotent translations on homogeneous spaces, Trans. Amer. Math.

Soc. 345 (1994), 577-594.

[21] R. J. Zimmer, Ergodic actions of semisimple groups and product relations, Annals of Mathematics, 118

(1983), 9-19.

[22] R. J. Zimmer, Ergodic Theory and Semisimple Groups, Birkhauser, 1984.

[23] R. J. Zimmer, Kazhdan groups acting on compact manifolds, Invent. Math. 75 (1984), 425-436.

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