scholarly journals Orbit equivalence and Borel reducibility rigidity for profinite actions with spectral gap

2016 ◽  
Vol 18 (12) ◽  
pp. 2733-2784 ◽  
Author(s):  
Adrian Ioana
Keyword(s):  
Spectral Gap ◽  
Orbit Equivalence ◽  
Borel Reducibility

scholarly journals Local Spectral Gap in the Group of Euclidean Isometries

2018 ◽  
Vol 2020 (2) ◽  
pp. 466-486
Author(s):  
Rémi Boutonnet ◽  
Adrian Ioana
Keyword(s):  
Euclidean Space ◽  
Spectral Gap ◽  
Main Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Orbit Equivalence ◽  
Dense Subgroup ◽  
Locally Compact ◽  
Strong Ergodicity

Abstract We provide new examples of translation actions on locally compact groups with the “local spectral gap property” introduced in [5]. This property has applications to strong ergodicity, the Banach–Ruziewicz problem, orbit equivalence rigidity, and equidecomposable sets. The main group of study here is the group $\operatorname{Isom}\left (\mathbb{R}^{d}\right )$ of orientation-preserving isometries of the Euclidean space $\mathbb{R}^{d}$, for d ≥ 3. We prove that the translation action of a countable dense subgroup Γ on Isom$\left (\mathbb R^{d}\right )$ has local spectral gap, whenever the translation action of the rotation projection of Γ on SO(d) has spectral gap. Our proof relies on the amenability of $\operatorname{Isom}\left (\mathbb{R}^{d}\right )$ and on work of Lindenstrauss and Varjú [12].

scholarly journals Ioana's Superrigidity Theorem and Orbit Equivalence Relations

ISRN Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Samuel Coskey
Keyword(s):  
Ergodic Theory ◽  
Set Theory ◽  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Borel Reducibility ◽  
Free Abelian Groups ◽  
Property T ◽  
Expository Paper ◽  
Torsion Free

We give a survey of Adrian Ioana's cocycle superrigidity theorem for profinite actions of Property (T) groups and its applications to ergodic theory and set theory in this expository paper. In addition to a statement and proof of Ioana's theorem, this paper features the following: (i) an introduction to rigidity, including a crash course in Borel cocycles and a summary of some of the best-known superrigidity theorems; (ii) some easy applications of superrigidity, both to ergodic theory (orbit equivalence) and set theory (Borel reducibility); and (iii) a streamlined proof of Simon Thomas's theorem that the classification of torsion-free abelian groups of finite rank is intractable.

PROPERTY τ AND COUNTABLE BOREL EQUIVALENCE RELATIONS

2007 ◽  
Vol 07 (01) ◽  
pp. 1-34 ◽  
Author(s):  
SIMON THOMAS
Keyword(s):  
Equivalence Relations ◽  
Orbit Equivalence ◽  
Borel Equivalence Relations ◽  
Borel Reducibility ◽  
Projective Lines

We prove Borel superrigidity results for suitably chosen actions of groups of the form SL2(ℤ[1/p1, … , 1/pt]), where {p1, …, pt} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL2(ℤ[1/q]) on the projective lines ℚp ∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals Free partition functions and an averaged holographic duality

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Nima Afkhami-Jeddi ◽  
Henry Cohn ◽  
Thomas Hartman ◽  
Amirhossein Tajdini
Keyword(s):  
Partition Function ◽  
Spectral Gap ◽  
Central Charge ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Partition Functions ◽  
General Constraints ◽  
Dimensional Gravity ◽  
Holographic Duality

Abstract We study the torus partition functions of free bosonic CFTs in two dimensions. Integrating over Narain moduli defines an ensemble-averaged free CFT. We calculate the averaged partition function and show that it can be reinterpreted as a sum over topologies in three dimensions. This result leads us to conjecture that an averaged free CFT in two dimensions is holographically dual to an exotic theory of three-dimensional gravity with U(1)c×U(1)c symmetry and a composite boundary graviton. Additionally, for small central charge c, we obtain general constraints on the spectral gap of free CFTs using the spinning modular bootstrap, construct examples of Narain compactifications with a large gap, and find an analytic bootstrap functional corresponding to a single self-dual boson.

scholarly journals Persistence of the spectral gap for the Landau–Pekar equations

2021 ◽  
Vol 111 (1) ◽  
Author(s):  
Dario Feliciangeli ◽  
Simone Rademacher ◽  
Robert Seiringer
Keyword(s):  
Initial Data ◽  
Spectral Gap ◽  
Effective Hamiltonian ◽  
Adiabatic Theorem ◽  
Strongly Coupled

AbstractThe Landau–Pekar equations describe the dynamics of a strongly coupled polaron. Here, we provide a class of initial data for which the associated effective Hamiltonian has a uniform spectral gap for all times. For such initial data, this allows us to extend the results on the adiabatic theorem for the Landau–Pekar equations and their derivation from the Fröhlich model obtained in previous works to larger times.

scholarly journals Nonzero spectral gap in several uniformly spin-2 and hybrid spin-1 and spin-2 AKLT models

2021 ◽  
Vol 3 (1) ◽  
Author(s):  
Wenhan Guo ◽  
Nicholas Pomata ◽  
Tzu-Chieh Wei
Keyword(s):  
Spectral Gap

scholarly journals On Polish groups admitting non-essentially countable actions

2020 ◽  
pp. 1-15
Author(s):  
ALEXANDER S. KECHRIS ◽  
MACIEJ MALICKI ◽  
ARISTOTELIS PANAGIOTOPOULOS ◽  
JOSEPH ZIELINSKI
Keyword(s):  
Equivalence Relation ◽  
Isometry Group ◽  
Alternative Proof ◽  
Orbit Equivalence ◽  
Locally Compact ◽  
Continuous Action ◽  
Polish Groups ◽  
Infinite Dimensional ◽  
Open Question ◽  
New Criterion

Abstract It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.

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