
A dichotomy for countable unions of smooth equivalence
relations.
With N. de Rancourt. To
appear in the Journal of Symbolic Logic.
We show that if an equivalence relation E on a Polish space is a countable union of smooth
Borel subequivalence relations, then there is either a Borel reduction of E to a countable
Borel equivalence relation on a Polish space or a continuous embedding of 𝔼_{1}
into E. We also establish related results concerning countable unions of more general
Borel equivalence relations.

The Feldman–Moore, Glimm–Effros, and Lusin–Novikov theorems over quotients.
With N. de Rancourt. To
appear in the Journal of Symbolic Logic.
We establish generalizations of the Feldman–Moore theorem, the Glimm–Effros
dichotomy, and the Lusin–Novikov uniformization theorem from Polish spaces to their
quotients by Borel orbit equivalence relations.

Markers and the ratio ergodic theorem.
With A. Tserunyan.
Ergodic Theory and Dynamical Systems: Proceedings of the Workshops University of North
Carolina at Chapel Hill 2021, 99116, 2024.
We establish a generalization and strengthening of the marker lemma for Borel automorphisms
that can also be viewed as a measureless strengthening of Dowker's ratio ergodic theorem.

A generalization of the 𝔾_{0} dichotomy and a strengthening of the
𝔼_{3} dichotomy.
Journal of Mathematical Logic, 2150028, 22 (1), 2022.
We generalize the 𝔾_{0} dichotomy to doublyindexed sequences of
analytic digraphs. Under a mild definability assumption, we use this generalization to
characterize the family of Borel actions of tsi Polish groups on Polish spaces that
can be decomposed into countablymany Borel actions admitting complete Borel sets that
are lacunary with respect to an open neighborhood of the identity. We also show that if
the group in question is nonarchimedean, then the inexistence of such a decomposition
yields a special kind of continuous embedding of 𝔼_{3} into the
corresponding orbit equivalence relation.

Scrambled Cantor sets.
With S. Geschke and
J. Grebík.
Proceedings of the American Mathematical Society, 44614468, 149 (10), 2021.
We show that Li–Yorke chaos ensures the existence of a scrambled Cantor set.

On the existence of small antichains for
definable quasiorders.
With
R. Carroy and Z. Vidnyánszky.
Journal of Mathematical Logic, 2150005, 21 (2), 2021.
We generalize Kada's definable strengthening of Dilworth's characterization of the class
of quasiorders admitting an antichain of a given finite cardinality.

Recurrence and the existence of invariant
measures.
With M. Inselmann. Journal of Symbolic Logic, 6076, 86 (1), 2021.
We show that recurrence conditions do not yield invariant Borel probability measures
in the descriptive settheoretic milieu, in the strong sense that if a Borel action
of a locally compact Polish group on a standard Borel space satisfies such a
condition but does not have an orbit supporting an invariant Borel probability
measure, then there is an invariant Borel set on which the action satisfies the
condition but does not have an invariant Borel probability measure.

Minimal definable graphs of definable chromatic
number at least three.
With
R. Carroy, D. Schrittesser, and
Z. Vidnyánszky. Forum of Mathematics, Sigma, E7 116, 9, 2021.
We show that there is a Borel graph on a standard Borel space of Borel chromatic
number three that admits a Borel homomorphism to every analytic graph on a standard
Borel space of Borel chromatic number at least three. Moreover, we characterize the
Borel graphs on standard Borel spaces of vertexdegree at most two with this property,
and show that the analogous result for digraphs fails.

On the existence of cocycleinvariant Borel
probability measures.
Ergodic Theory and Dynamical Systems, 31503168, 40 (11), 2020.
We show that a natural generalization of compressibility is the sole obstruction
to the existence of a cocycleinvariant Borel probability measure.

Bases for functions beyond the first
Baire class.
With
R. Carroy. Journal of Symbolic Logic, 12891303, 85 (3), 2020.
We provide a finite basis for the class of Borel functions that are not in the first Baire
class, as well as the class of Borel functions that are not σcontinuous with closed
witnesses.

The open dihypergraph dichotomy and
the second level of the Borel hierarchy.
With
R. Carroy and
D. Soukup. Contemporary Mathematics, 119, 752, 2020.
We show that several dichotomy theorems concerning the second level of the Borel hierarchy
are special cases of the ℵ_{0}dimensional generalization of the open graph
dichotomy, which itself follows from the usual proof(s) of the perfect set theorem. Under
the axiom of determinacy, we obtain the generalizations of these results from analytic to
separable metric spaces. We also consider connections between cardinal invariants and the
chromatic numbers of the corresponding dihypergraphs.

On the existence of large antichains for definable
quasiorders.
With Z. Vidnyánszky.
Journal of Symbolic Logic, 103108, 85 (1), 2020.
We simultaneously generalize Silver's perfect set theorem for coanalytic
equivalence relations and Harrington–Marker–Shelah's Dilworthstyle
perfect set theorem for Borel quasiorders, establish the analogous theorem at
the next definable cardinal, and give further generalizations under weaker
definability conditions.

Incomparable actions of free groups.
With C. Conley. Ergodic
Theory and Dynamical Systems, 20842098, 37 (7), 2017.
Suppose that X is a Polish space, E is a countable Borel equivalence relation
on X, and μ is an Einvariant Borel probability measure on X. We consider
the circumstances under which for every countable nonabelian free group
Γ, there is a Borel sequence (·_{r})_{r∈ℝ}
of free actions of Γ on X, generating subequivalence relations E_{r}
of E with respect to which μ is ergodic, with the further property that
(E_{r})_{r∈ℝ} is an increasing sequence of relations which
are pairwise incomparable under μreducibility. In particular, we show that if
E satisfies a natural separability condition, then this is the case as long as
there exists a free Borel action of a countable nonabelian free group on X,
generating a subequivalence relation of E with respect to which μ is ergodic.

Sigmacontinuity with closed witnesses.
With
R. Carroy. Fundamenta Mathematicae, 2942, 239, 2017.
We use variants of the 𝔾_{0} dichotomy to establish a refinement of
Solecki's basis theorem for the family of Baireclass one functions which are not
σcontinuous with closed witnesses.

Measurable perfect matchings for acyclic locally
countable Borel graphs.
With C. Conley. Journal of Symbolic
Logic, 258271, 82 (1), 2017.
We characterize the structural impediments to the existence of Borel perfect
matchings for acyclic locally countable Borel graphs admitting a Borel selection of
finitely many ends from their connected components. In particular, this yields the
existence of Borel matchings for such graphs of degree at least three. As a corollary,
it follows that acyclic locally countable Borel graphs of degree at least three
generating μhyperfinite equivalence relations admit μmeasurable matchings.
We establish the analogous result for Baire measurable matchings in the locally finite
case, and provide a counterexample in the locally countable case.

Dichotomy theorems for families of noncofinal
essential complexity.
With J. Clemens and D. Lecomte.
Advances in Mathematics, 285299, 304, 2017.
We prove that for every Borel equivalence relation E, either E is Borel reducible
to 𝔼_{0}, or the family of Borel equivalence relations incompatible
with E has cofinal essential complexity. It follows that if F is a Borel equivalence
relation and ℱ is a family of Borel equivalence relations of noncofinal essential
complexity which together satisfy the dichotomy that for every Borel equivalence
relation E, either E ∈ ℱ or F is Borel reducible to E, then ℱ consists
solely of smooth equivalence relations, thus the dichotomy is equivalent to a known
theorem.

Orthogonal measures and strong ergodicity.
With C. Conley. Israel Journal of
Mathematics, 8389, 218 (1), 2017.
Burgess–Mauldin have proven the Ramseytheoretic result that continuous
sequences (μ_{c})_{c∈2ℕ} of pairwise orthogonal
Borel probability measures admit continuous orthogonal subsequences. We establish an
analogous result for sequences indexed by 2^{ℕ}/𝔼_{0},
the next Borel cardinal. As a corollary, we obtain a strengthening of the
Harrington–Kechris–Louveau 𝔼_{0} dichotomy for restrictions
of measure equivalence. We then use this to characterize the family of countable Borel
equivalence relations which are nonhyperfinite with respect to an ergodic Borel
probability measure that is not strongly ergodic.

Measure reducibility of countable Borel
equivalence relations.
With C. Conley. Annals of
Mathematics, 347402, 185 (2), 2017.
(And a correction.)
We show that every basis for the countable Borel equivalence relations strictly
above 𝔼_{0} under measure reducibility is uncountable, thereby
ruling out natural generalizations of the Glimm–Effros dichotomy. We also
push many known results concerning the abstract structure of the measure
reducibility hierarchy to its base, using arguments substantially simpler than
those previously employed.

A bound on measurable chromatic
numbers of locally finite Borel graphs.
With C. Conley. Mathematical
Research Letters, 16331644, 23 (6), 2016.
We show that the Baire measurable chromatic number of every locally finite Borel
graph on a nonempty Polish space is strictly less than twice its ordinary chromatic
number, provided this ordinary chromatic number is finite. In the special case that
the connectedness equivalence relation is hyperfinite, we obtain the analogous
result for the μmeasurable chromatic number.

The smooth ideal.
With C. Conley and J. Clemens.
Proceedings of the London Mathematical Society, 5780, 112 (1), 2016.
We give a classical proof of the generalization of the characterization of
smoothness to quotients of Polish spaces by Borel equivalence relations. As an
application, we describe the extent to which any given Borel equivalence relation
on a Polish space is encoded by the corresponding σideal generated by the
family of Borel sets on which it is smooth.

An embedding theorem of 𝔼_{0}
with modeltheoretic applications.
With I. Kaplan. Journal of
Mathematical Logic, 1450010, 14 (2), 2014.
We provide a new criterion for embedding 𝔼_{0} and apply it to equivalence
relations in model theory. This generalizes the results of the authors and Pierre
Simon on the Borel cardinality of Lascar strong types equality, and Newelski’s
results about pseudo F_{σ} groups.

The Borel cardinality of Lascar strong
types.
With I. Kaplan and
P. Simon. Journal of the London
Mathematical Society, 609630, 90 (2), 2014.
We show that if the restriction of the Lascar equivalence relation to a KPstrong
type is nontrivial, then it is nonsmooth (when viewed as a Borel equivalence
elation on an appropriate space of types).

Essential countability of treeable
equivalence relations.
With J. Clemens and D. Lecomte.
Advances in Mathematics, 131, 265, 2014.
We establish a dichotomy theorem characterizing the circumstances under which a
treeable Borel equivalence relation E is essentially countable. Under additional
topological assumptions on the treeing, we in fact show that E is essentially
countable if and only if there is no continuous embedding of 𝔼_{1} into
E. Our techniques also yield the first classical proof of the analogous result
for hypersmooth equivalence relations, and allow us to show that, up to continuous
Kakutani embeddability, there is a minimum Borel function which is not essentially
countabletoone.

An antibasis result for graphs
of infinite Borel chromatic number.
With C. Conley. Proceedings
of the American Mathematical Society, 21232133, 142 (6), 2014.
We answer in the negative a question posed by Kechris–Solecki–Todorcevic
as to whether the shift graph on Baire space is minimal among graphs of
indecomposably infinite Borel chromatic number. To do so, we use ergodictheoretic
techniques to construct a new graph amalgamating various properties of the shift
actions of free groups. The resulting graph is incomparable with any graph induced
by a function. We then generalize this construction and collect some of its useful
properties.

Stationary probability measures and
topological realizations.
With C. Conley and
A. Kechris. Israel Journal
of Mathematics, 333345, 198 (1), 2013.
We establish the generic inexistence of stationary Borel probability measures
for aperiodic Borel actions of countable groups on Polish spaces. Using this, we
show that every aperiodic continuous action of a countable group on a compact
Polish space has an invariant Borel set on which it has no σcompact
realization.

Incomparable treeable equivalence
relations.
Journal of Mathematical Logic, 121162, 12 (1), 2012.
We establish Hjorth's theorem that there is a family of continuummany
pairwise strongly incomparable free actions of free groups, and therefore a
family of continuummany pairwise incomparable treeable equivalence relations.

The graphtheoretic approach to
descriptive set theory.
Bulletin of Symbolic Logic, 554575, 18 (4), 2012.
We sketch the ideas behind the use of chromatic numbers in establishing
descriptive settheoretic dichotomy theorems.

Definability of small puncture sets.
With A. Caicedo, J. Clemens,
and C. Conley. Fundamenta
Mathematicae, 3951, 215 (1), 2011.
We characterize the class of definable families of countable sets for which
there is a single countable definable set intersecting every element of the
family.

Dichotomy theorems for countably
infinite dimensional analytic hypergraphs.
Annals of Pure and Applied Logic, 561565, 162 (7), 2011.
We give classical proofs, strengthenings, and generalizations of Lecomte's
characterizations of analytic ωdimensional hypergraphs with countable
Borel chromatic number.

A classical proof of the
Kanovei–Zapletal canonization.
Contemporary Mathematics, 281285, 533, 2011.
We give a classical proof of the Kanovei–Zapletal canonization of
Borel equivalence relations on Polish spaces.

Descriptive Kakutani equivalence.
With C.
Rosendal. Journal of the European Mathematical Society,
179219, 12 (1), 2010.
We consider a descriptive settheoretic analog of Kakutani equivalence
for Borel automorphisms of Polish spaces. Answering a question of
Nadkarni, we show that up to this notion, there are exactly two aperiodic
Borel automorphisms of uncountable Polish spaces. Using this, we classify all
Borel ℝflows up to C^{∞}timechange isomorphism. We then
extend the notion of descriptive Kakutani equivalence to all (not necessarily
injective) Borel functions, and provide a variety of results leading towards
a complete classification. The main technical tools are a series of
Glimm–Effros and Dougherty–Jackson–Kechrisstyle embedding
theorems.

Ends of graphed equivalence relations, II.
With G. Hjorth. Israel Journal of Mathematics, 393415, 169, 2009.
Given a graphing G of a countable Borel equivalence relation on a Polish
space, we show that if there is a Borel way of selecting a nonempty closed set
of countably many ends from each Gcomponent, then there is a Borel way of
selecting an end or line from each Gcomponent. Our method yields also
Glimm–Effros style dichotomies which characterize the circumstances
under which: (1) there is a Borel way of selecting a point or end from each
Gcomponent, and (2) there is a Borel way of selecting a point, end, or line
from each Gcomponent.

Ends of graphed equivalence relations, I.
With G. Hjorth. Israel Journal of Mathematics, 375392, 169, 2009.
Given a countable Borel equivalence relation E on a Polish space, we show: (1)
E admits an endless graphing if and only if E is smooth, (2) E admits a locally
finite singleended graphing if and only if E is aperiodic, (3) E admits a
graphing for which there is a Borel way of selecting two ends from each component
if and only if E is hyperfinite, and (4) E admits a graphing for which there is a
Borel way of selecting a finite set of at least three ends from each component if
and only if E is smooth.

Basis theorems for nonpotentially closed
sets and graphs of uncountable Borel chromatic number.
With D. Lecomte. Journal
of Mathematical Logic, 121162, 8 (2), 2008.
We show that there is an antichain basis for neither (1) the class of
nonpotentially closed Borel subsets of the plane under Borel rectangular
reducibility nor (2) the class of analytic graphs of uncountable Borel chromatic
number under Borel reducibility.

Measurable chromatic numbers.
Journal of Symbolic Logic, 11391157, 73 (4), 2008.
We show that if add(null) = 𝔠, then the globally Baire and universally
measurable chromatic numbers of the graph of any Borel function on a Polish
space are equal and at most three. In particular, this holds for the graph
of the unilateral shift on [ℕ]^{ℕ}, although its Borel
chromatic number is ℵ_{0}. We also show that if add(null) = 𝔠,
then the universally measurable chromatic number of every treeing of a measure
amenable equivalence relation is at most three. In particular, this holds for
the minimum analytic graph 𝔾_{0} with uncountable Borel (and Baire
measurable) chromatic number. In contrast, we show that for all κ ∈
{2, 3, . . . , ℵ_{0}, 𝔠}, there is a treeing of 𝔼_{0}
with Borel and Baire measurable chromatic number κ. Finally, we use a
Glimm–Effros style dichotomy theorem to show that every basis for a
nonempty initial segment of the class of graphs of Borel functions of Borel
chromatic number at least three contains a copy of (ℝ^{<ℕ}, ⊇).

The existence of
quasiinvariant measures of a given cocycle, II: Probability measures.
Ergodic Theory and Dynamical Systems, 16151633, 28 (5), 2008.
Given a Polish space X, a countable Borel equivalence relation E on X, and
a Borel cocycle ρ : E → (0, ∞), we characterize the circumstances
under which there is a probability measure μ on X such that ρ(φ^{1}(x), x)
= [d(φ∗μ)/dμ](x) μalmost everywhere, for every Borel injection
φ whose graph is contained in E.

The existence of quasiinvariant
measures of a given cocycle, I: Atomless, ergodic sigmafinite measures.
Ergodic Theory and Dynamical Systems, 15991613, 28 (5), 2008.
Given a Polish space X, a countable Borel equivalence relation E on X, and a
Borel cocycle ρ : E → (0, ∞), we characterize the circumstances
under which there is a suitably nontrivial σfinite measure μ on X
such that ρ(φ^{1}(x), x) = [d(φ∗μ)/dμ](x) μalmost
everywhere, for every Borel injection φ whose graph is contained in E.

Means on equivalence relations.
With A. Kechris.
Israel Journal of Mathematics, 241263, 163, 2008.
Suppose that X is a Polish space and E is a countable Borel equivalence
relation on X. We show that if there is a Borel assignment of means to
the equivalence classes of E, then E is smooth. We also show that if there
is a Baire measurable assignment of means to the equivalence classes of E,
then E is generically smooth.

Coordinatewise decomposition of
groupvalued Borel functions.
Fundamenta Mathematicae, 119126, 196 (2), 2007.
Answering a question of Kłopotowski–Nadkarni–Sarbadhikari–Srivastava,
we characterize the Borel sets S ⊆ X × Y on which every Borel function f :
S → ℂ is of the form uvS, where u : X → ℂ and v : Y → ℂ
are Borel.

Isomorphism of Borel full groups.
With C. Rosendal.
Proceedings of the American Mathematical Society, 517522, 135 (2), 2007.
Suppose that G and H are Polish groups which act in a Borel fashion on Polish spaces X
and Y. Let E_{G} and E_{H} denote the
corresponding orbit equivalence relations, and [G] and [H] the corresponding Borel full
groups. Modulo the obvious counterexamples, we show that [G] ≅ [H] ⇔ E_{G}
≅_{B} E_{H}.

Coordinatewise decomposition, Borel
cohomology, and invariant measures.
Fundamenta Mathematicae, 8194, 191 (1), 2006.
Given Polish spaces X and Y and a Borel set S ⊆ X × Y with
countable sections, we describe the circumstances under which a Borel function f : S
→ ℝ is of the form f(x, y) = u(x) + v(y), where u : X → ℝ and v : Y
→ ℝ are Borel. This turns out to be a special case of the problem of
determining whether a realvalued Borel cocycle on a countable Borel equivalence
relation is a coboundary. We use several Glimm–Effros style dichotomies to give a
solution to this problem in terms of certain σfinite measures on the underlying
space. The main new technical ingredient is a characterization of the existence of type
III measures of a given cocycle.

Full groups, classification, and
equivalence relations.
PhD Dissertation, 281 pages, 2004.
In Chapter I, we study algebraic properties of full groups of automorphisms of
σcomplete Boolean algebras. We consider problems of writing automorphisms as
compositions of periodic automorphisms and commutators (generalizing work of Fathi
and Ryzhikov), as well as problems concerning the connection between normal subgroups
of a full group and ideals on the underlying algebra, in the process giving a new proof
(joint with David Fremlin) of Shortt's characterization of the normal subgroups of the
group of Borel automorphisms of an uncountable Polish space, as well as a characterization
of the normal subgroups of full groups of countable Borel equivalence relations which are
closed in the uniform topology of Bezuglyi–Dooley–Kwiatkowski. We also
characterize the existence of an Einvariant Borel probability measure in terms of a
purely algebraic property of [E]. The results of Chapter II include classifications of
Borel automorphisms and Borel forests of lines up to the descriptive analog of Kakutani
equivalence, along with applications to the study of Borel marriage problems, generalizing
and strengthening results of Shelah–Weiss, Dougherty–Jackson–Kechris,
and Kłopotowski–Nadkarni–Sarbadhikari–Srivastava. We also study the sorts
of full groups on quotients of the form X/E for which the results of Chapter I do not apply.
Actions of such groups satisfy a measureless ergodicity property which we exploit to obtain
various classification and rigidity results. In particular, we obtain descriptive analogs of
some results of Connes–Krieger and Feldman–Sutherland–Zimmer, answering
a question of Bezuglyi. In Chapter III, we study some descriptive properties of quasiinvariant
measures. We prove a general selection theorem, and use this to show a descriptive settheoretic
strengthening of an analog of the Hurewicz ergodic theorem which holds for all countable Borel
equivalence relations. This then leads to new proofs of Ditzen’s quasiinvariant ergodic
decomposition theorem and Nadkarni’s characterization of the existence of an Einvariant
probability measure, and also gives rise to a quasiinvariant version of Nadkarni’s theorem,
as well as a version for countabletoone Borel functions. We close chapter III with results on
graphings of countable Borel equivalence relations, strengthening theorems of Adams and Paulin.

Topics in orbit
equivalence.
With A. Kechris.
Lecture Notes in Mathematics, 1852, Springer, 2004.
This volume provides a selfcontained introduction to some topics in orbit
equivalence theory, a branch of ergodic theory. The first two chapters focus
on hyperfiniteness and amenability. Included here are proofs of Dye's theorem
that probability measurepreserving, ergodic actions of the integers are orbit
equivalent and of the theorem of Connes–Feldman–Weiss identifying amenability
and hyperfiniteness for nonsingular equivalence relations. The presentation
here is often influenced by descriptive set theory, and Borel and generic analogs
of various results are discussed. The final chapter is a detailed account of
Gaboriau's recent results on the theory of costs for equivalence relations and
groups and its applications to proving rigidity theorems for actions of free groups.