PROJECT DESCRIPTION
Robin Tucker-Drob
1. B
My primary interests lie at the interface of descriptive set theory, ergodic theory, representation theory,
and measurable group theory. In studying problems in measurable group theory and the ergodic theory
and representation theory of countable groups, I have been aided and often motivated by a global perspective informed by descriptive set theory. From this perspective, problems in ergodic theory may be seen as
topological-dynamical and descriptive problems concerning actions of the Polish group A = A(X, µ) of automorphisms of a standard (usually non-atomic) probability space (X, µ). Likewise, representation theory
may be studied via actions of the Polish group U(H) of unitary operators on a separable (usually infinite
dimensional) Hilbert space H.
The proposed research revolves around several open problems, each having ergodic theoretic and representation theoretic aspects whose relationship is poignantly perceived from this global viewpoint. These
problems show great promise for fruitful and compelling interactions among the above-mentioned fields.
1.1. Approximation in the ergodic theory of countable groups. A (probability-)measure preserving action of
a discrete countably infinite group Γ on (X, µ) is a homomorphism a ∶ Γ → A(X, µ). The set of all measure
preserving actions of Γ on (X, µ) naturally forms a Polish space A(Γ, X, µ) on which A acts continuously by
coordinate-wise conjugation. The orbit A⋅a of a ∈ A(Γ, X, µ) is called its conjugacy class and two actions a and
b from A(Γ, X, µ) with the same conjugacy class are said to be conjugate. We say that b is weakly contained
in a if it is in the closure of the conjugacy class of a, and we call a and b weakly equivalent if each weakly
contains the other. If a ∈ A(Γ, X, µ) and b ∈ A(Γ, Y, ν) are actions with different underlying probabilities
spaces then we say that b is weakly contained in a if it is a factor (i.e., quotient) of some c ∈ A(Γ, X, µ) that
is weakly equivalent to a. Weak containment of measure preserving actions was introduced by Kechris in
[35] as an ergodic theoretic analogue of weak containment of unitary representations, and it has proven
to be a remarkably robust notion that accurately captures an intuition that one measure preserving action
asymptotically approximates or simulates another. Abért and Elek have recently defined a compact Polish
topology on the set of weak equivalence classes in which many important invariants of weak equivalence
become continuous functions [2], [47]. A fundamental theorem regarding weak containment is due to Abért
and Weiss and concerns the Bernoulli shift action of Γ which we now define.
Let Γ act on the set [0, 1]Γ of functions f ∶ Γ → [0, 1] by shifting indices: (γ ⋅ f )(δ) = f (γ −1 δ). This action
preserves the product measure ν Γ where ν is Lebesgue measure, and we call this measure preserving action
the Bernoulli shift of Γ and denoted it by sΓ . The Bernoulli shift provides an ergodic theoretic counterpart to
the left regular representation of Γ.
Theorem 1.1 (Abért-Weiss [4]). sΓ is weakly contained in every free measure preserving action of Γ.
Conversely, any measure preserving action weakly containing sΓ must itself be free. A measure preserving action is therefore free if and only if it exhibits approximate Bernoulli behavior. Adrian Ioana conjectured that there is in fact an absorption principle at work which strengthens this.
Conjecture 1.2 (A. Ioana). Let a be any free measure preserving action of a countably infinite group Γ. Then sΓ × a
is weakly contained in a.
Conjecture 1.2 strengthens Theorem 1.1 since the product action sΓ × a is easily seen to weakly contain
each of its factors. By combining ideas from [3] with a close analysis of weak containment I showed that an
even more general absorption principle holds, of which Ioana's conjecture is a special case [47].
Theorem 1.3. Conjecture 1.2 is true.
Theorem 1.3 provided a means of answering a question of Abért and Elek regarding the relationship
between conjugacy and weak equivalence.
Question 1.4 (Abért-Elek [2]). Does there exist a countably infinite group Γ with a free measure preserving
action whose conjugacy class and weak equivalence class coincide?
1
2
This question may be addressed within the descriptive set theoretic framework of Borel reducibility,
which provides a means of comparing the relative set theoretic complexity of equivalence relations naturally arising in mathematics. The answer to Question 1.4 is given by Theorem 1.5 below, following a brief
introduction to this framework.
1.2. Borel reducibility and classification. If E and F are equivalence relations on standard Borel spaces
X and Y respectively, then E is called Borel reducible to F , denoted E ≤B F , if there is a Borel map ψ ∶
X → Y satisfying xEy ⇔ ψ(x)F ψ(y) for all x, y ∈ X. Such a map ψ is called a Borel reduction from E to
F . The substance of this notion lies in the requirement that this map be definable in some sense, and there
are theoretical reasons for choosing Borel definability. The resulting richness of the ordering ≤B and its
continuing success in comparing naturally occurring equivalence relations in mathematics may be taken as
further justifications for this choice. A Borel reduction from E to F may be seen as providing an explicitly
definable classification of elements of X up to E-equivalence using the F -classes as invariants.
An equivalence relation is said to be classifiable by countable structures if it is Borel reducible to the isomorphism relation on some standard Borel space of countable structures, for example, countable graphs,
groups, or partial orders. More precisely, E admits classification by countable structures if there exists a
countable language L and a Borel reduction from E to isomorphism on the standard Borel space XL of all
L-structures with universe N. A classical example of such a classification is the Halmos-von Neumann Theorem which completely classifies all ergodic measure preserving transformations with discrete spectrum,
up to isomorphism, by their group of eigenvalues [32]. Another example is Ellio 's complete classification
of unital AF-algebras by their pointed pre-ordered K0 -groups [20], [22]. On the other hand, Hjorth has
isolated a dynamical property called turbulence that may hold of a Polish group action, and which is an
obstruction to there being a classification by countable structures for the orbit equivalence relation of that
action (in fact, turbulence is in a sense the only obstruction [30]). Combining Theorem 1.3 with the work of
Kerr, Li, and Pichot [37] on turbulence in spaces of C ∗ -algebra representations, I proved the following:
Theorem 1.5. Let a be any free measure preserving action of a countably infinite group Γ. Then the conjugacy relation
on the weak equivalence class of a is not classifiable by countable structures.
This implies that the weak equivalence class of a contains a continuum of conjugacy classes, and thus
answers Question 1.4. But the conclusion is actually much stronger than this: there is no Borel way of
assigning countable trees, rings, orderings etc., as invariants to actions in the weak equivalence class of a
that completely classifies these actions up to conjugacy.
1.3. Automatic Freeness. The subject of non-free measure preserving actions has received significant a ention recently, see for example [3, 12, 13, 15, 19, 47, 48, 49, 51, 1, 2, 25, 50, 10, 45]. In [45], Stuck and Zimmer
prove a strong generalization of the Margulis Normal Subgroup Theorem for certain higher-rank semisimple Lie groups in terms of an automatic freeness property for many measure preserving actions of these
groups. One consequence is that if Γ is an irreducible la ice in such a group then any non-atomic ergodic
a ∈ A(Γ, X, µ) is almost free, i.e., there exists a finite normal subgroup N of Γ such that the stabilizer Γx of
almost every x ∈ X is equal to N . This is an example of automatic freeness at one extreme: by restricting
considerably the group Γ, a minimal hypothesis on the action is needed to ensure that it is almost free. Recently, I proved an automatic freeness result at the other extreme in which Γ is only assumed infinite, but
a more serious ergodicity assumption is imposed on the action [48]. A measure preserving action of Γ is
called totally ergodic if each infinite subgroup of Γ acts ergodically and it is called trivial if the underlying
measure is a point mass. I showed the following:
Theorem 1.6. All non-trivial totally ergodic actions of countably infinite groups are almost free. In particular, all
non-trivial mixing actions and all non-trivial mildly mixing actions of countably infinite groups are almost free.
This is new even for the case of mixing actions; Weiss had previously observed that actions of amenable
groups with a much stronger mixing property called completely positive entropy are almost free. The total
ergodicity assumption is close to optimal since there are examples due to Vershik [51] of actions with mixing
properties only slightly weaker than mild mixing, but which are totally non-free, which means that these
examples are in some sense as far from free as possible. The most surprising aspect of Theorem 1.6 is that
its proof uses the group theoretic fact that every infinite locally finite group contains an infinite abelian
3
subgroup, and all known proofs of this fact in turn rely on the Feit-Thompson odd order theorem from
finite group theory [33, 26, 44].
My subsequent investigations into natural analogues of Theorem 1.6 have revealed connections to well
known open questions about group C ∗ -algebras as well as to the theory of cost, all discussed below.
2. P
R
2.1. Amenable Invariant Random Subgroups. The freeness properties of an action a ∈ A(Γ, X, µ) may be
studied directly via that action's stabilizer distribution, obtained as the image of the measure µ under the
stabilizer map x ↦ Γx . This defines a Borel probability measure on the space of subgroups of Γ that is
invariant under conjugation by elements of Γ. Any such probability measure is called an invariant random
subgroup of Γ, so-named by Abért, Glasner, and Virag, who showed that every invariant random subgroup
of Γ arises as the stabilizer distribution of some measure preserving action of Γ [3]. Each normal subgroup
of Γ is an invariant random subgroup when viewed as a Dirac distribution and many theorems originally
concerning normal subgroups have been shown to generalize to invariant random subgroups, the StuckZimmer Theorem being one prominent example. In what follows, an invariant random subgroup of Γ will
be said to have a particular property if it has that property with probability 1.
Open Question 2.1. Is every amenable invariant random subgroup of a countable group Γ contained in
some amenable normal subgroup of Γ?
While this is open in general, Y. Glasner [24] has obtained a positive answer for linear groups (see also the
remark after (Diagram 0)). There is a useful way of restating Question 2.1 in terms of the amenable radical
of a group. Day showed that every discrete group Γ contains a characteristic subgroup, called the amenable
radical of Γ and denoted ARΓ , which is amenable and which contains all other amenable normal subgroups
of Γ. Question 2.1 is then equivalent to the question of whether a countable group with trivial amenable
radical has no non-trivial amenable invariant random subgroups.
2.2. Shift-minimality and C ∗ -simplicity. If C is a class of groups then a measure preserving action of a
group Γ is called C-ergodic if each subgroup of Γ in C acts ergodically. Using an idea from the proof of
Theorem 1.6, I showed that if a non-trivial action of Γ is NA-ergodic, where NA is the class of non-amenable
groups, then the invariant random subgroup associated to this action is amenable. One may show that
every measure preserving action weakly contained in the Bernoulli shift sΓ is NA-ergodic, and therefore
any non-trivial action weakly contained in sΓ gives rise to an amenable invariant random subgroup of Γ
which will be non-trivial provided the original action was not free. Call a countable group Γ shift-minimal
if every non-trivial action weakly contained in sΓ is free.
Open Question 2.2. If the amenable radical of Γ is trivial then is Γ shift-minimal?
The Abért-Weiss characterization of free actions as those weakly containing sΓ yields that Γ is shiftminimal if and only if every non-trivial action weakly contained in sΓ is in fact weakly equivalent to sΓ .
It is well known that Γ is C ∗ -simple, i.e., the reduced C ∗ -algebra, Cr∗ (Γ), of Γ is simple, if and only if every non-zero unitary representation of Γ weakly contained in the left-regular representation λΓ is actually
weakly equivalent to λΓ [27]. This is a tantalizing parallel, although there is no obvious implication between
the two properties.
Open Question 2.3. Are all C ∗ -simple groups shift-minimal?
C ∗ -simplicity may be restated as a dynamical property of an action of the unitary group U(H), where
H = ℓ2 (Γ). The set Irrλ (Γ, H) of all irreducible representations of Γ on H weakly contained in λΓ naturally
forms a Polish space on which U(H) acts continuously by coordinate-wise conjugation. Then Γ is C ∗ -simple
if and only if Γ is ICC and every unitary conjugacy class in Irrλ (Γ, H) is dense.
Evidence suggests that C ∗ -simple groups should be shift-minimal. I have shown for example that shiftminimality of Γ follows from another property called the unique trace property, which means that Cr∗ (Γ) has
a unique tracial state, and the proof of this indicates an approach to answering Question 2.3 positively. In
all known examples, the unique trace property and C ∗ -simplicity coincide, although it is open whether this
is the case in general.
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Theorem 2.4. Groups with the unique trace property are shift-minimal. In fact, groups with the unique trace property
have no non-trivial amenable invariant random subgroups.
Powers [42] demonstrated C ∗ -simplicity and the unique trace property for non-abelian free groups, and
since then many large classes of groups have been shown to have both of these properties [28, 11, 7, 8, 6,
27, 29]. It is notable that in many cases, including the original argument of Powers, the proof given for a
group's C ∗ -simplicity makes use of stronger hypotheses than the corresponding proof that the group has
the unique trace property. The following diagram depicts the known implications among the five notions
discussed. Any implication not addressed by the diagram is an open problem in general.
(Diagram 0)
C ∗ -simple
No non-trivial ks
amenable IRS
TD
TD
Unique trace
Shift-minimal mu
TD
TD
[40]
'/ Trivial amenable nv
[40]
radical
Theorem 2.4 and results of Poznansky [43] imply these properties are all equivalent for linear groups.
2.3. Cost and the first ℓ2 -Be i number. Another interesting development is a connection between shiftminimality and cost. The cost of a measure preserving countable Borel equivalence relation is a [0, ∞]-valued
orbit equivalence invariant introduced by Levi [39] and then developed considerably by Gaboriau [23]. The
cost of a measure preserving action of Γ is defined to be the cost of the equivalence relation generated by this
action. The cost of a group Γ, denoted C(Γ), is then defined as the infimum of the costs of its free measure
preserving actions. When Γ is infinite, then C(Γ) ≥ 1. Γ is said to have fixed price r, where r ≥ 0, if every free
action of Γ has cost r. For example, infinite amenable groups have fixed price 1, and Gaboriau has shown
the free group of rank n has fixed price n. A major open question in the area is whether every countable
group has fixed price. This is known to be the case for many groups, but is open in general. Recently, I
proved the following, providing many examples of shift-minimal groups:
Theorem 2.5. If a countable group Γ does not have fixed price 1 then Γ/ARΓ is shift-minimal. In addition, if C(Γ) > 1
then every non-trivial invariant random subgroup of Γ/ARΓ of infinite index has cost ∞, and in particular Γ/ARΓ
has no non-trivial amenable invariant random subgroups.
Results of Gaboriau imply ARΓ is finite in the above situation. Part of the proof of the first statement in
Theorem 2.5 involves extending results of Kechris [35] on cost and weak containment in finitely generated
groups to general countable groups; a consequence is that a countable group has fixed price 1 if and only if
its Bernoulli shift has cost 1. The second statement is an analogue of a theorem of Bergeron and Gaboriau
[10] about the first ℓ2 -Be i number. Theorems 2.4 and 2.5 along with their result provide strong evidence
for a positive answer to the following question:
Open Question 2.6. Let Γ be a countably infinite group and assume the first ℓ2 -Be i number of Γ is positive.
Must Γ/ARΓ be C ∗ -simple and have the unique trace property?
(2)
(2)
It is known that C(Γ) ≥ β1 (Γ) + 1 for any countably infinite group Γ, where β1 (Γ) is the first ℓ2 -Be i
(2)
number of Γ. It is an open problem whether this is actually an equality. Regardless, the hypothesis β1 (Γ) >
0 is at least as strong as the hypothesis C(Γ) > 1 from Theorem 2.5. Peterson and Thom [41] have shown
that if Γ is torsion-free and satisfies an additional technical hypothesis, then Questions 2.6 has a positive
answer. What they actually show is that groups satisfying their hypotheses have many free subgroups, and
then C ∗ -simplicity and the unique trace property are easily deduced using a Powers-like argument from
[8]. If the additional technical hypothesis is dropped then their methods still show that Γ has rather strong
paradoxicality properties which may still be enough to deduce C ∗ -simplicity or the unique trace property.
This would unify and generalize many known examples of groups with these properties. It would also
provide additional evidence for a positive answer to Question 2.3, and it would add an interesting dimension
to the relationship between cost and the first ℓ2 -Be i number.
5
3. A
3.1. Borel complexity of group representations. Let ≈Γ denote the unitary conjugacy equivalence relation
associated to the conjugation action of U(H) on the Polish space Rep(Γ, H) of representations of Γ on an
infinite dimensional separable Hilbert space H. Let ̂
≈Γ denote the restriction of ≈Γ to the Polish subspace
Irr(Γ, H) of irreducible representations. Hjorth has shown that if Γ is not abelian-by-finite, then ̂
≈Γ is not
classifiable by countable structures [31] (see also [37]). If Γ and ∆ are both amenable groups neither of which
is abelian-by-finite, then Thomas has shown, drawing on [46, 21], that ̂
≈Γ and ̂
≈∆ are Borel bi-reducible, i.e.,
̂
≈Γ ≤B ̂
≈∆ and ̂
≈ ∆ ≤B ̂
≈Γ . This is one of the few Borel reducibility results comparing equivalence relations
which are not classifiable by countable structures. It is striking that, although the statement is purely representation theoretic, the proof uses methods and results from ergodic theory, most notably the OrnsteinWeiss Theorem. Thomas has conjectured the following:
Open Conjecture 3.1 (S. Thomas). Let F2 denote the free group on two generators. Then
(1) ≈F2 ≰B ≈Z
(2) If Γ is any amenable group then ̂
≈F2 ≰B ̂
≈Γ .
3.2. Properties MD and FD. A residually finite group Γ has property MD [34] if the finite actions (i.e., actions coming from finite quotients of Γ) are dense in A(Γ, X, µ), and Γ has FD [38] if the finite representations
are dense in Rep(Γ, H). It is not difficult to show that MD implies FD, but the converse is unknown. It is
known that free groups and residually finite amenable groups have MD [34], MD is closed under taking
subgroups, and I have shown MD is closed under taking free products [47]. The groups SLn (Z) for n ≥ 3
are known to not have FD [38] and hence do not have MD. Lewis Bowen and I [14] have shown another
closure property of MD which implies that surface groups have MD and -- in light of the recent proof of the
Virtual Fibration Conjecture [5] -- that fundamental groups of closed hyperbolic 3-manifolds have property
MD. The natural next step is to study the following possible generalization:
Open Question 3.2 (L. Bowen). Do hyperbolic groups which act properly and cocompactly on CAT(0) cube
complexes have property MD?
3.3. Further collaborations. I am working with Miklos Abért on the question of whether there are uncountably many weak equivalence classes of free ergodic actions in A(Γ, X, µ) whenever Γ contains a non-abelian
free subgroup or, more generally, when Γ is nonamenable. I also have an ongoing project with Clinton Conley and Alexander Kechris examining measurable combinatorial invariants of measure preserving actions
[16]. One of the main problems here is to determined the exact √
value of the measurable independence num1
2m−1
] by [17].
ber of sFm . It is known to lie somewhere in the interval [ 2m
, m+√
2m−1
4. C
An NSF postdoctoral research fellowship is exactly what I need at this point in my career. It would
provide me with the means and flexibility to most effectively advance the proposed research and foster
current collaborations, as well as engage in new ones. To give a tangible example, I have been invited, and
plan to a end, the workshop "Group Theory, Measure, and Asymptotic Invariants" at Oberwolfach next
summer, and the research allowance granted by this fellowship would help offset the expenses of this trip.
Simon Thomas is a prominent leader in the field of descriptive set theory and he has done significant
research in many fields including group theory and ergodic theory. He has been an influential presence in
my mathematical upbringing through both his work and the many conversations we have had at conferences and workshops, and I count him as one of my mentors. All this makes Simon the ideal choice for my
sponsoring scientist during this postdoctoral fellowship. Additionally, Rutgers is a hub for descriptive set
theory and logic. The logic group hosts many visitors, has a weekly descriptive set theory seminar, a weekly
logic seminar, and hosts the MAMLS meetings each year bringing together many people in the field.
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[2] M. Abért and G. Elek, The space of actions, partition metric, and combinatorial rigidity, arXiv:1108.2147.
[3] M. Abért, Y. Glasner, and B. Virag, Kesten's Theorem for Invariant Random Subgroups, arXiv:1201.3399.
[4] M. Abért and B. Weiss, Bernoulli actions are weakly contained in any free action. arXiv:1103.1063.
[5] I. Agol, with an appendix by I. Agol, D. Groves, and J. Manning, The virtual Haken conjecture, arXiv:1204.2810.
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[9] B. Bekka and P. de la Harpe, Groups with Simple Reduced C ∗ -algebras. Expo. Math. 18 (2000), no. 3, 215-230.
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[13] L. Bowen, R. Grigorchuk, and R. Kravchenko, Invariant random subgroups of the lamplighter group, arXiv:1206.6780.
[14] L. Bowen and R. Tucker-Drob, On a co-induction question of Kechris, Israel J. of Math., to appear. doi: 10.1007/s11856-0120071-7 (available at arXiv:1105.0648).
[15] D. Creu and J. Peterson, Stabilizers of Ergodic Actions of La ices and Commensurators, preprint (2012).
[16] C. Conley, A. Kechris, and R. Tucker-Drob, Ultraproducts of measure preserving actions and graph combinatorics, Ergodic Theory
and Dynam. Systems, Available on CJO 2012 doi:10.1017/S0143385711001143.
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[47]
[48]
[49]
[50]
[51]
R. Tucker-Drob, Weak equivalence and non-classifiability of measure preserving actions, arXiv:1202.3101v2.
R. Tucker-Drob, Mixing actions of countable groups are almost free, arXiv:1208.0655.
R. Tucker-Drob, Shift-minimal groups, preprent.
A. Vershik, Nonfree actions of countable groups and their characters, J. Math. Sci. (N. Y.) 174 (2011), no. 1, 1-6.
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