We establish a new characterization of property (T) in terms of the Furstenberg entropy of nonsingular actions. Given any generating measure μ on a countable group G, A. Nevo showed that a necessary condition for G to have property (T) is that the Furstenberg μ-entropy values of the ergodic, properly nonsingular G-actions are bounded away from zero. We show that this is also a sufficient condition

Bowen, Lewis

Hartman, Yair

Tamuz, Omer

Caltech Authors - Main

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Property (T) and the Furstenberg Entropy of
Nonsingular Actions
Lewis Bowen∗, Yair Hartman†and Omer Tamuz‡
July 1, 2014
Abstract
We establish a new characterization of property (T) in terms of the Furstenberg
entropy of nonsingular actions. Given any generating measure µ on a countable group
G, A. Nevo showed that a necessary condition for G to have property (T) is that
the Furstenberg µ-entropy values of the ergodic, properly nonsingular G-actions are
bounded away from zero. We show that this is also a sufficient condition.
1 Introduction
A measurable action on a probability space Gy(X, η) is called a nonsingular action if the
measure η is quasi-invariant with respect to any g ∈ G; that is, if η and g∗η are equivalent (i.e.,
mutually absolutely continuous) measures. We say that the action is properly nonsingular if
η is not equivalent to a G-invariant probability measure.
A probability measure µ on a countable group G is generating if its support generates G
as a semigroup.
Let Gy(X, η) be a nonsingular action and let µ be a probability measure on G. The
Furstenberg entropy [5] or µ-entropy is defined by
hµ(X, η) =
∑
g∈G
µ(g)
∫
X
− log
dg−1∗ η
dη
(x) dη(x).
Jensen’s inequality implies that hµ(X, η) ≥ 0 and that, for generating measures, equality
holds if and only if η is an invariant measure. Furstenberg entropy is an important conjugacy
invariant of nonsingular actions [11], and in particular of stationary actions; Gy(X, η) is
µ-stationary if µ ∗ η = η.
∗University of Texas at Austin. Supported in part by NSF grant DMS-0968762, NSF CAREER Award
DMS-0954606 and BSF grant 2008274.
†Weizmann Institute of Science. Supported by the European Research Council, grant 239885.
‡Microsoft Research.
1
The pair (G, µ) is said to have an entropy gap if there exists some constant ǫ = ǫ(G, µ) > 0
such that the µ-entropy of any ergodic, properly nonsingular G-action on a probability space
is at least ǫ.
A group G has property (T), if any unitary representation of G which has almost invariant
vectors admits a non-zero invariant vector. The purpose of this paper is to establish a similar
characterization of property (T), using nonsingular actions instead of unitary representations,
and thinking of those with small entropy values as being “almost invariant”.
Theorem 1. Let G be a countable group. Then G has property (T) if and only if for every
(equivalently, for some) generating measure µ, (G, µ) has an entropy gap.
For the case that G has property (T) this theorem follows from Nevo [10, Propositions 4.1,
4.2]. Note that the statements of these propositions omit the requirements that η is ergodic
and properly nonsingular; the result is no longer true without either of these hypotheses.
For completeness, we include in Appendix A a reproduction of Nevo’s proof of this direction.
The idea of the proof of the other direction (i.e., that groups without property (T) have
no entropy gap) is the following. We first consider the group 2Nfin; the elements of this
group are the finite subsets of N and the group operation is symmetric difference. For this
group, we construct in Section 2 very simple nonsingular actions which show that 2Nfin has
no entropy gap, for a family of natural measures on 2Nfin.
Then, in Section 3, we observe that any countable group G without property (T) admits a
non-trivial cocycle into 2Nfin. Using these cocycles one can construct skew-product G-actions
over 2Nfin spaces; the entropy formula for skew-product actions is established in Section 4.
Finally, in Section 5 we prove Theorem 1, by showing that the 2Nfin-actions that we construct
in Section 2, and which have arbitrarily low Furstenberg entropy, can be lifted to skew-
product G-actions with arbitrarily low Furstenberg entropy.
1.1 Related literature
1.1.1 Characterizing property (T)
Recently, Ozawa [12] showed that property (T) can be characterized by the spectral gap
of ∆µ ∈ R[G], the Laplacian associated with a finitely supported, symmetric generating
measure µ. This characterization is one of many which have been established since Kazhdan’s
first definition of property (T) [9], some of which have been instrumental in proving property
(T) for groups that were previously not known to have it. We refer the reader to Bekka, de
La Harpe and Valette [1] for a complete discussion.
1.1.2 Stationary actions
A particularly interesting class of nonsingular actions are the stationary ones. Furstenberg
entropy has been a useful tool in the study of stationary actions (e.g., [5, 8]), and the study of
the set of entropy values realizable by properly nonsingular stationary actions has attracted
some interest [11, 2, 6].
2
Nevo’s theorem (Theorem A.1) implies that for every (G, µ), where G has property (T),
the Furstenberg entropy values of properly nonsingular, ergodic µ-stationary actions are
bounded away from zero; in this case we say that (G, µ) has a stationary entropy gap. In [2]
and [6] it is shown that some (G, µ) without property (T) have no stationary entropy gap.
However, this is not a characterization of property (T): in a previous paper [3, Proposition
7.5] we show that there exist (G, µ) without property (T), but with a stationary entropy
gap.
A question that therefore remains open is that of characterizing the pairs (G, µ) that have
a stationary entropy gap. More narrowly, it is not even known that all amenable groups have
no stationary entropy gap.
2 The case of 2Nfin
Let 2Nfin denote the set of all finite subsets of the natural numbers N = {1, 2, 3, . . .}. Endowed
with the operation of symmetric difference, 2Nfin is an abelian group in which every element
other than the identity is an involution. For T ∈ 2Nfin, let max(T ) = maxt∈T t. We also let
max(∅) = 0. Because 2Nfin is an abelian group we will use additive notation when expressing
group multiplication. Thus T + S := T △ S = (T \ S) ∪ (S \ T ) for any T, S ∈ 2Nfin.
Let 2Nfin act on 2
N by symmetric difference, identifying elements of 2N with subsets of N.
Let ωp be a measure on 2
N given by ωp =
∏
N
B(p), where B(p) is the Bernoulli p measure
on {0, 1}.
Proposition 2.1. Consider the action 2Nfiny(2
N, ωp). Then, for any probability measure µ
on 2Nfin, the µ-entropy of (2
N, ωp) is
hµ(2
N, ωp) = ϕ(p)
∑
T∈2N
fin
µ(T )|T | ≤ ϕ(p)
∑
T∈2N
fin
µ(T )max(T ),
where ϕ(p) = log p
1−p
(2p− 1).
Note that limp→1/2 ϕ(p) = 0 and that for p 6= 1/2 it holds that ϕ(p) > 0.
Proof. For T ∈ 2Nfin, T
−1
∗ ωp = T∗ωp =
∏
n∈T B(1− p)
∏
n 6∈T B(p). Hence∫
X
− log
dT∗ω
dω
(x) dω(x) = ϕ(p)|T |
where ϕ(p) = log p
1−p
(2p− 1) is the Kullback-Leibler divergence between B(p) and B(1− p).
Finally, it follows that for any measure µ on 2Nfin,
hµ(2
N, ωp) = ϕ(p)
∑
T∈2N
fin
µ(T )|T | ≤ ϕ(p)
∑
T∈2N
fin
µ(T )max(T ).
3
It is easy to show that ωp is ergodic and is not equivalent to an invariant probability
measure (see Lemma 4.2; an invariant measure λ would have to satisfy λ({x : xn = 1}) =
1/2). Hence it follows from this proposition that 2Nfin does not have an entropy gap, for any
µ with
∑
T∈2N
fin
µ(T )max(T ) < ∞. In the remainder of this paper we lift this result to any
group without property (T).
3 Non-property (T) and cocycles
Before stating the next proposition, let us recall some standard definitions. Given a nonsin-
gular action Gy(X, η) and a countable group Γ, a cocycle c : G ×X → Γ is a measurable
map such that
c(gh, x) = c(g, hx)c(h, x)
for a.e. x and every g, h ∈ G. Two such cocycles c1, c2 : G × X → Γ are cohomologous if
there exists a measurable function β : X → Γ such that
c1(g, x) = β(gx)c2(g, x)β(x)
−1
for every g ∈ G and a.e. x. A cocycle is a coboundary if it is cohomologous to the trivial
cocycle (whose essential image is contained in the trivial element {e}).
Proposition 3.1. Assume G does not have property (T). Let ρ : G → N be a proper
function (so ρ−1(n) is finite for every n ∈ N). Then there exists an ergodic probability
measure preserving action Gy(X, η) and a Borel cocycle c : G × X → 2Nfin such that c is
not cohomologous to a cocycle with a finite image. Moreover, for every element g ∈ G,∫
X
max(c(g, x)) dη(x) ≤ ρ(g).
Proof. This result follows from the proof of [7, Theorem 2.1] as we now explain. It is a well-
known result of Connes and Weiss [4] that if G does not have property (T) then there exists
an ergodic probability measure preserving action Gy(X, η) which is not strongly ergodic
(see also [1, Theorem 6.3.4]). By [7, Lemma 2.4] there exist Borel sets Dn ⊂ X such that
the following hold.
1. Dn is asymptotically invariant: limn→∞ η(gDn △ Dn) = 0 for every g ∈ G;
2. limn→∞ η(Dn) = 1/2.
3. For every g ∈ G with ρ(g) ≤ n, η(Dn △ gDn) < 2
−n.
To be precise, the last statement follows from an easy modification of the proof of [7, Lemma
2.4]; see equation (2.10) there.
Define a map φ : X → 2N by φ(x) = {j ∈ N : x ∈ Dj}. We let 2
N
fin act on 2
N by
symmetric difference.
4
Let g ∈ G. If m ≥ ρ(g) then
η({x ∈ X : φ(gx) /∈ 2Nfin △ φ(x)}) ≤
∞∑
k=m
η(Dk △ gDk) < 2
−m+1.
Sincem ≥ ρ(g) is arbitrary, the left hand side must equal zero. Therefore φ(gx) ∈ 2Nfin △ φ(x)
for η-a.e. x ∈ X .
Define the cocycle c : G × X → 2Nfin by c(g, x) = T if φ(gx) = T △ φ(x). If ρ(g) = m
then ∫
X
max(c(g, x)) dη(x) ≤ m− 1 +
∫
{x:max(c(g,x))≥m}
max(c(g, x)) dη(x)
≤ m− 1 +
∞∑
n=m
η({x ∈ X : max(c(g, x)) ≥ n})
≤ m− 1 +
∞∑
n=m
η(Dn △ gDn) ≤ m− 1 + 2
−m+1 ≤ m.
The cocycle c cannot be cohomologous to a cocycle with a finite image, since φ is non-
trivial. To be precise, for m ∈ N, we let 2m < 2Nfin denote the subgroup consisting of all
elements T ∈ 2Nfin with support in [m] := {1, . . . , m}. To obtain a contradiction, suppose
there is a Borel map f : X → 2Nfin and a cocycle b : G×X → 2
m such that
c(g, x) = f(gx) + b(g, x) + f(x)
for a.e. x ∈ X and every g ∈ G. Note that all elements of 2Nfin are involutions, and so
−f(x) = f(x).
Let ψ(x) = f(x) △ φ(x). Observe that
ψ(gx) = f(gx) △ φ(gx) = f(gx) △ c(g, x) △ φ(x) = f(gx) △ f(gx) △ b(g, x) △ f(x) △ φ(x)
= b(g, x) △ ψ(x)
for a.e. x ∈ X and every g ∈ G.
Let ψ˜(x) = ψ(x)\ [m], f˜(x) = f(x)\ [m] and φ˜(x) = φ(x)\ [m]. Then, since b(g, x) ⊆ [m],
ψ˜(gx) = (b(g, x) △ ψ(x)) \ [m] = ψ˜(x),
and ψ˜ is G-invariant. Because Gy(X, η) is ergodic, there is an element R ∈ 2N\[m] such that
ψ˜(x) = R for a.e. x. Thus
R = ψ˜(x) = f˜(x) △ φ˜(x),
which implies φ˜(x) △ R = f˜(x) ∈ 2Nfin for a.e. x ∈ X .
Let D′n = Dn if n 6∈ R, and otherwise let D
′
n = X \Dn, the complement of Dn. Then, for
a.e. x ∈ X , there are only a finite number of elements j ∈ N \ [m] such that x ∈ D′j; these
are precisely the elements in f˜(x).
It follows that if we let An = {x ∈ X : max(f˜(x)) ≥ n}, then limn η(An) = 0. But An
contains D′n, and so η(An) ≥ η(D
′
n), in contradiction to the fact that limn η(D
′
n) = 1/2.
5
4 Entropy for skew-products
Lemma 4.1. Let G be a countable group with a probability measure µ. Let Gy(X, η) be
a probability measure preserving action, and let c : G × X → Γ be a cocycle to a countable
group Γ. Also, let Γy(W,ω) be a nonsingular action on the probability space (W,ω). Let
GyX ×W be the skew-product action
g(x, w) = (gx, c(g, x)w).
Then
hµ(X ×W, η × ω) =
∫
X
hµx(W,ω) dη(x)
where µx is the pushforward of µ under the map g 7→ c(g, x).
Proof.
hµ(X ×W, η × ω) = −
∑
g∈G
µ(g)
∫
X×W
log
(
dg−1∗ (η × ω)
dη × ω
(x, w)
)
dη(x)dω(w)
= −
∑
g∈G
µ(g)
∫
X×W
log
(
dc(g, x)−1∗ ω
dω
(w)
)
dη(x)dω(w)
= −
∫
X
∑
γ∈Γ
µx(γ)
∫
W
log
(
dγ−1∗ ω
dω
(w)
)
dω(w)dη(x)
=
∫
X
hµx(W,ω) dη(x).
Lemma 4.2. Let En = {x ∈ 2
N : xn = 1}. Let ν be a Borel probability measure on 2
N such
that ν(En) ≤ 1/2 for infinitely many n ∈ N, and let ωp be the i.i.d. Bernoulli p measure on
2N, with p > 1/2. Then ν is not absolutely continuous with respect to ωp.
Proof. Let {ik}
∞
k=1 be a sequence of indices such that ν(Eik) < 1/2. For x ∈ 2
N, denote
Sn(x) =
1
n
n∑
k=1
1Eik (x),
and let
S(x) = lim inf
n
Sn(x),
so that, clearly, ωp({x ∈ 2
N : S(x) 6= p}) = 0. But∫
X
Sn(x) dν(x) =
1
n
n∑
k=1
ν(Eik) ≤ 1/2,
6
and so by Fatou’s Lemma,
∫
X
S(x) dν(x) ≤ lim inf
n
1
n
n∑
k=1
ν(Eik) ≤ 1/2 < p.
Hence ν({x ∈ 2N : S(x) 6= p}) > 0, and ν is not absolutely continuous with respect to
ωp.
Lemma 4.3. Let Gy(X, η) be an ergodic probability measure preserving action, let c :
G × X → 2Nfin be a cocycle, and let 2
N
fin act on 2
N by symmetric difference. Consider the
skew-product action GyX × 2N given by g(x, y) = (gx, c(g, x)y).
Denote by ωp the Bernoulli p i.i.d. measure on 2
N. For any 1/2 < p < 1, if there exists
a G-invariant probability measure that is absolutely continuous with respect to η × ωp, then
c is cohomologous to a cocycle with a finite image.
Proof. Let λ ≪ η × ωp be a G-invariant probability measure. Note that λ projects to a
G-invariant probability measure on X which is absolutely continuous with respect to η.
Because η is ergodic, this projection must equal η. Hence
dλ(x, y) = dη(x)dλx(y)
where λ =
∫
δx × λx dη(x) is the disintegration of λ over η (here δx is the Dirac measure
concentrated on {x}). Also
dλ
dη × ωp
(x, y) =
dλx
dωp
(y).
It follows that λx ≪ ωp, for a.e. x. By the invariance of λ we have that, for every g ∈ G,
g∗λ = λ which implies
λgx = (g∗λ)gx = c(g, x)∗λx
for a.e. x.
Let En = {x ∈ 2
N : xn = 1}, and let f(x) = {n ∈ N : λx(En) = 1/2}. By Lemma 4.2
f(x) is finite for a.e. x ∈ X , since λx ≪ ωp for a.e. x ∈ X . Then
f(gx) = {n ∈ N : λgx(En) = 1/2}
= {n ∈ N : c(g, x)∗λx(En) = 1/2}
= f(x),
since for any measure ν on 2N and T ∈ 2Nfin, T∗ν(En) = 1 − ν(En) if n ∈ T and T∗ν(En) =
ν(En) otherwise; in any case, T∗ν(En) = 1/2 iff ν(En) = 1/2.
By the ergodicity of η it follows that there exists a T ∈ 2Nfin such that f(x) = T for a.e.
x ∈ X . Let m = max(T ), and denote c˜(g, x) = c(g, x) \ {1, . . . , m}.
7
Let ψ(x) = {n > m : λx(En) < 1/2}. Then
ψ(gx) = {n > m : λgx(En) < 1/2}
= {n > m : c(g, x)∗λx(En) < 1/2}
= c˜(g, x) △ {n > m : λx(En) < 1/2}
= c˜(g, x) △ ψ(x),
where the third equality is a consequence of the fact that λx(En) 6= 1/2 for every n > m
and a.e. x ∈ X . Note that ψ(x) is finite, by another application of Lemma 4.2. Hence c˜ is a
coboundary, and so c is cohomologous to the cocycle c △ c˜ whose image is in 2m.
5 Proof of main theorem
Proof of Theorem 1. By Theorem A.1 we may assume G does not have property (T). Let µ
be a generating measure on G and let ρ : G→ N be a proper function such that∑
g∈G
µ(g)ρ(g) ≤ 2.
By Proposition 3.1 there exists an ergodic probability measure preserving action Gy(X, η)
and a cocycle c : G ×X → 2Nfin that is not cohomologous to a cocycle with a finite image.
Moreover, ∫
X
max(c(g, x)) dη(x) ≤ ρ(g)
for any g ∈ G. By Proposition 2.1 for every ǫ > 0 there exists a Bernoulli probability
measure ωp(ǫ) on 2
N such that for any probability measure ν on 2Nfin,
hν(2
N, ωp(ǫ)) ≤ ǫ
∑
T∈2N
fin
ν(T )max(T ).
Consider the skew-product action GyX × 2N given by g(x, T ) = (gx, c(g, x)T ). By
Lemma 4.1,
hµ(X × 2
N, η × ωp(ǫ)) =
∫
X
hµx(2
N, ωp(ǫ)) dη(x)
≤ ǫ
∫
X
∑
T∈2N
fin
µx(T )max(T ) dη(x).
Note that, by Proposition 3.1, we know that∫
X
∑
T∈2N
fin
µx(T )max(T ) dη(x) =
∑
g∈G
µ(g)
∫
X
max(c(g, x)) dη(x) ≤
∑
g∈G
µ(g)ρ(g) ≤ 2,
8
and so
lim
ǫ→0
hµ(X × 2
N, η × ωp(ǫ)) = 0.
Since c is not cohomologous to a cocycle with a finite image, Lemma 4.3 implies that
each measure η× ωp(ǫ) is properly nonsingular, and furthermore almost every measure in its
ergodic decomposition is properly nonsingular. Since the entropy of η × ωp(ǫ) is a convex
combination of the entropies of its ergodic components, it follows that there exist ergodic,
properly nonsingular G-actions with arbitrarily small entropy, and therefore (G, µ) does not
have an entropy gap.
A Groups with property (T)
Theorem A.1 (Nevo). Let G be a countable group with property (T), and let µ be a gener-
ating measure. Then (G, µ) has an entropy gap.
The following proof is based on Nevo’s [10].
Proof. Let µ¯ be the measure on G given by
µ¯ =
∞∑
n=0
2−n−1µn,
where µn is the convolution of µ with itself n times. Then hµ¯(X, η) = hµ(X, η) (see, e.g. [8],
or [6, Section 2.8]). The advantage of µ¯ is that it is supported everywhere on G.
Consider, given a nonsingular action Gy(X, η), the unitary representation π on L2(X, η)
given by
π(g)f(x) =
√
dg∗η
dη
(x)f(g−1x).
Note that π(g)∗ = π(g−1).
It is easy to check that by Jensen’s inequality, for any g ∈ G,
−2 log 〈1, π(g)1〉 = −2 log
〈
π(g−1)1, 1
〉
≤
∫
X
− log
dg−1∗ η
dη
(x)dη(x). (1)
Consider the Markov operator π(µ¯) : L2(X, η) → L2(X, η) given by π(µ¯) =
∑
g∈G µ¯(g)π(g).
Then (1) yields the bound −2 log ‖π(µ¯)‖ ≤ hµ¯(X, η), by another application of Jensen’s
inequality.
Denote
‖µ¯‖T = sup {‖π (µ¯) ‖ : π is a unitary representation of G with no invariant vectors} .
If G has property (T) then ‖µ¯‖T < 1 (see, e.g., Bekka, de La Harpe and Valette [1, Corollary
6.2.3]); here we use the fact that µ¯ is supported everywhere.
When Gy(X, η) is properly nonsingular ergodic action, L2(X, η) has no invariant vectors
(see [3, Lemma 7.2]). It therefore follows that
0 < −2 log ‖µ¯‖T ≤ −2 log ‖π(µ¯)‖ ≤ hµ¯(X, η) = hµ(X, η),
and (G, µ) has an entropy gap, with ǫ(µ) = −2 log ‖µ¯‖T .
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10


Property (T) and the Furstenberg Entropy of Nonsingular Actions

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