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We show that if a hyperbolic group acts geometrically on a CAT(0) cube complex, then the induced boundary action is hyperfinite. This means that for a cubulated hyperbolic group, the natural action on its Gromov boundary is hyperfinite, which generalizes an old result of Dougherty, Jackson and Kechris for the free group case.
We derive a forcing axiom from the conjunction of square and diamond, and present a few applications, primary among them being the existence of super-Souslin trees. It follows that for every uncountable cardinal
$\unicode[STIX]{x1D706}$
, if
$\unicode[STIX]{x1D706}^{++}$
is not a Mahlo cardinal in Gödel’s constructible universe, then
$2^{\unicode[STIX]{x1D706}}=\unicode[STIX]{x1D706}^{+}$
entails the existence of a
$\unicode[STIX]{x1D706}^{+}$
-complete
$\unicode[STIX]{x1D706}^{++}$
-Souslin tree.
In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.
If
$3\leqslant n<\unicode[STIX]{x1D714}$
and
$V$
is a vector space over
$\mathbb{Q}$
with
$|V|\leqslant \aleph _{n-2}$
, then there is a well ordering of
$V$
such that every vector is the last element of only finitely many length-
$n$
arithmetic progressions (
$n$
-APs). This implies that there is a set mapping
$f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$
with no free set which is an
$n$
-AP. If, however,
$|V|\geqslant \aleph _{n-1}$
, then for every set mapping
$f:V\rightarrow [V]^{{<}\unicode[STIX]{x1D714}}$
there is a free set which is an
$n$
-AP.
The purpose of this paper twofold. Firstly, we establish
$\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness and
$\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness of several different classes of multifractal decomposition sets of arbitrary Borel measures (satisfying a mild non-degeneracy condition and two mild “smoothness” conditions). Secondly, we apply these results to study the
$\unicode[STIX]{x1D6F1}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness and
$\unicode[STIX]{x1D6F4}_{\unicode[STIX]{x1D6FE}}^{0}$
-completeness of several multifractal decomposition sets of self-similar measures (satisfying a mild separation condition). For example, a corollary of our results shows if
$\unicode[STIX]{x1D707}$
is a self-similar measure satisfying the strong separation condition and
$\unicode[STIX]{x1D707}$
is not equal to the normalized Hausdorff measure on its support, then the classical multifractal decomposition sets of
$\unicode[STIX]{x1D707}$
defined by
We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the groups of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. This answers a question of Ben Yaacov, Berenstein and Melleray. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the automatic continuity property for the group
$\text{Aut}([0,1],\unicode[STIX]{x1D706})$
, due to Ben Yaacov, Berenstein and Melleray and for the unitary group of the infinite-dimensional separable Hilbert space, due to Tsankov.
We study the relationship between a
$\unicode[STIX]{x1D705}$
-Souslin tree
$T$
and its reduced powers
$T^{\unicode[STIX]{x1D703}}/{\mathcal{U}}$
.
Previous works addressed this problem from the viewpoint of a single power
$\unicode[STIX]{x1D703}$
, whereas here, tools are developed for controlling different powers simultaneously. As a sample corollary, we obtain the consistency of an
$\aleph _{6}$
-Souslin tree
$T$
and a sequence of uniform ultrafilters
$\langle {\mathcal{U}}_{n}\mid n<6\rangle$
such that
$T^{\aleph _{n}}/{\mathcal{U}}_{n}$
is
$\aleph _{6}$
-Aronszajn if and only if
$n<6$
is not a prime number.
This paper is the first application of the microscopic approach to Souslin-tree construction, recently introduced by the authors. A major component here is devising a method for constructing trees with a prescribed combination of freeness degree and ascent-path characteristics.
We introduce the open degree of a compact space, and we show that for every natural number
$n$
, the separable Rosenthal compact spaces of degree
$n$
have a finite basis.
Forster [‘Finite-to-one maps’, J. Symbolic Logic68 (2003), 1251–1253] showed, in Zermelo–Fraenkel set theory, that if there is a finite-to-one map from
${\mathcal{P}}(A)$
, the set of all subsets of a set
$A$
, onto
$A$
, then
$A$
must be finite. If we assume the axiom of choice (AC), the cardinalities of
${\mathcal{P}}(A)$
and the set
$S(A)$
of permutations on
$A$
are equal for any infinite set
$A$
. In the absence of AC, we cannot make any conclusion about the relationship between the two cardinalities for an arbitrary infinite set. In this paper, we give a condition that makes Forster’s theorem, with
${\mathcal{P}}(A)$
replaced by
$S(A)$
, provable without AC.
The classical theorem of Vizing states that every graph of maximum degree
$d$
admits an edge coloring with at most
$d+1$
colors. Furthermore, as it was earlier shown by Kőnig,
$d$
colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree
$d$
admits a measurable edge coloring with
$d+O(\sqrt{d})$
colors; furthermore, if the graphing has no odd cycles, then
$d+1$
colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that
$d+1$
colors are always enough.
We generalize Brooks’ theorem to show that if
$G$
is a Borel graph on a standard Borel space
$X$
of degree bounded by
$d\geqslant 3$
which contains no
$(d+1)$
-cliques, then
$G$
admits a
${\it\mu}$
-measurable
$d$
-coloring with respect to any Borel probability measure
${\it\mu}$
on
$X$
, and a Baire measurable
$d$
-coloring with respect to any compatible Polish topology on
$X$
. The proof of this theorem uses a new technique for constructing one-ended spanning subforests of Borel graphs, as well as ideas from the study of list colorings. We apply the theorem to graphs arising from group actions to obtain factor of IID
$d$
-colorings of Cayley graphs of degree
$d$
, except in two exceptional cases.
The Weyl–von Neumann theorem asserts that two bounded self-adjoint operators A, B on a Hilbert space H are unitarily equivalent modulo compacts, i.e.uAu* + K = B for some unitary u 𝜖 u(H) and compact self-adjoint operator K, if and only if A and B have the same essential spectrum: σess (A) = σess (B). We study, using methods from descriptive set theory, the problem of whether the above Weyl–von Neumann result can be extended to unbounded operators. We show that if H is separable infinite dimensional, the relation of unitary equivalence modulo compacts for bounded self-adjoint operators is smooth, while the same equivalence relation for general self-adjoint operators contains a dense Gδ-orbit but does not admit classification by countable structures. On the other hand, the apparently related equivalence relation A ~ B ⇔ ∃u 𝜖 U(H) [u(A-i)–1u* - (B-i)–1 is compact] is shown to be smooth.
We prove, in
$\text{ZF}+\boldsymbol{{\it\Sigma}}_{2}^{1}$
-determinacy, that, for any analytic equivalence relation
$E$
, the following three statements are equivalent: (1)
$E$
does not have perfectly many classes, (2)
$E$
satisfies hyperarithmetic-is-recursive on a cone, and (3) relative to some oracle, for every equivalence class
$[Y]_{E}$
we have that a real
$X$
computes a member of the equivalence class if and only if
${\it\omega}_{1}^{X}\geqslant {\it\omega}_{1}^{[Y]}$
. We also show that the implication from (1) to (2) is equivalent to the existence of sharps over
$ZF$
.
In this note we partially answer a question of Cascales, Orihuela and Tkachuk [‘Domination by second countable spaces and Lindelöf
${\rm\Sigma}$
-property’, Topology Appl.158(2) (2011), 204–214] by proving that under
$CH$
a compact space
$X$
is metrisable provided
$X^{2}\setminus {\rm\Delta}$
can be covered by a family of compact sets
$\{K_{f}:f\in {\it\omega}^{{\it\omega}}\}$
such that
$K_{f}\subset K_{h}$
whenever
$f\leq h$
coordinatewise.
We study countable saturation of metric reduced products and introduce continuous fields of metric structures indexed by locally compact, separable, completely metrizable spaces. Saturation of the reduced product depends both on the underlying index space and the model. By using the Gelfand–Naimark duality we conclude that the assertion that the Stone–Čech remainder of the half-line has only trivial automorphisms is independent from ZFC (Zermelo-Fraenkel axiomatization of set theory with the Axiom of Choice). Consistency of this statement follows from the Proper Forcing Axiom, and this is the first known example of a connected space with this property.
Given a cardinal
$\lambda $
with
$\lambda =\lambda ^{\aleph _0}$
, we show that there is a field of cardinality
$\lambda $
whose automorphism group is a free group of rank
$2^\lambda $
. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank
$2^\lambda $
is the automorphism group of a field of cardinality
$\lambda $
whenever
$\lambda $
is a cardinal with
$\lambda =\lambda ^{\aleph _0}$
. Moreover, we apply these techniques to show that consistently the assumption that
$\lambda =\lambda ^{\aleph _0}$
is not necessary for the existence of a field of cardinality
$\lambda $
whose automorphism group is a free group of rank
$2^\lambda $
. Finally, we use them to prove that the existence of a cardinal
$\lambda $
of uncountable cofinality with the property that there is no field of cardinality
$\lambda $
whose automorphism group is a free group of rank greater than
$\lambda $
implies the existence of large cardinals in certain inner models of set theory.
It has been a long-standing question whether every amenable operator algebra is isomorphic to a (necessarily nuclear)
$\mathrm{C}^*$
-algebra. In this note, we give a nonseparable counterexample. Finding out whether a separable counterexample exists remains an open problem. We also initiate a general study of unitarizability of representations of amenable groups in
$\mathrm{C}^*$
-algebras and show that our method cannot produce a separable counterexample.
We study the consequences of stationary and semi-stationary set reflection. We show
that the semi-stationary reflection principle implies the Singular Cardinal
Hypothesis, the failure of the weak square principle, etc. We also consider two
cardinal tree properties introduced recently by Weiss, and prove that they follow
from stationary and semi-stationary set reflection augmented with a weak form of
Martin’s Axiom. We also show that there are some differences between the two
reflection principles, which suggests that stationary set reflection is analogous to
supercompactness, whereas semi-stationary set reflection is analogous to strong
compactness.