scholarly journals HIGH DENSITY PIECEWISE SYNDETICITY OF PRODUCT SETS IN AMENABLE GROUPS

2016 ◽  
Vol 81 (4) ◽  
pp. 1555-1562 ◽  
Author(s):  
MAURO DI NASSO ◽  
ISAAC GOLDBRING ◽  
RENLING JIN ◽  
STEVEN LETH ◽  
MARTINO LUPINI ◽  
...  
Keyword(s):  
Lower Bound ◽  
Finite Subset ◽  
Amenable Group ◽  
High Density ◽  
Amenable Groups ◽  
Quantitative Version ◽  
Banach Density ◽  
Piecewise Syndetic ◽  
Product Sets

AbstractM. Beiglböck, V. Bergelson, and A. Fish proved that if G is a countable amenable group and A and B are subsets of G with positive Banach density, then the product set AB is piecewise syndetic. This means that there is a finite subset E of G such that EAB is thick, that is, EAB contains translates of any finite subset of G. When G = ℤ, this was first proven by R. Jin. We prove a quantitative version of the aforementioned result by providing a lower bound on the density (with respect to a Følner sequence) of the set of witnesses to the thickness of EAB. When G = ℤd, this result was first proven by the current set of authors using completely different techniques.

Tilings of amenable groups

2019 ◽  
Vol 2019 (747) ◽  
pp. 277-298 ◽  
Author(s):  
Tomasz Downarowicz ◽  
Dawid Huczek ◽  
Guohua Zhang
Keyword(s):  
Fixed Points ◽  
Topological Entropy ◽  
Finite Subset ◽  
Dimensional Space ◽  
Amenable Group ◽  
Free Action ◽  
Amenable Groups ◽  
Entropy Zero

Abstract We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

scholarly journals Definable sets containing productsets in expansions of groups

2019 ◽  
Vol 22 (1) ◽  
pp. 63-82
Author(s):  
Uri Andrews ◽  
Gabriel Conant ◽  
Isaac Goldbring
Keyword(s):  
Recent Work ◽  
Amenable Group ◽  
Amenable Groups ◽  
First Order ◽  
Definable Subset ◽  
Banach Density ◽  
Definable Sets ◽  
Infinite Sets

Abstract We consider the question of when sets definable in first-order expansions of groups contain the product of two infinite sets (we refer to this as the “productset property”). We first show that the productset property holds for any definable subset A of an expansion of a discrete amenable group such that A has positive Banach density and the formula {x\cdot y\in A} is stable. For arbitrary expansions of groups, we consider a “1-sided” version of the productset property, which is characterized in various ways using coheir independence. For stable groups, the productset property is equivalent to this 1-sided version, and behaves as a notion of largeness for definable sets, which can be characterized by a natural weakening of model-theoretic genericity. Finally, we use recent work on regularity lemmas in distal theories to prove a definable version of the productset property for sets of positive Banach density definable in certain distal expansions of amenable groups.

scholarly journals Lower bound in the Roth theorem for amenable groups

2014 ◽  
Vol 35 (6) ◽  
pp. 1746-1766 ◽  
Author(s):  
QING CHU ◽  
PAVEL ZORIN-KRANICH
Keyword(s):  
Lower Bound ◽  
Probability Measure ◽  
Amenable Group ◽  
Amenable Groups ◽  
Measurable Set ◽  
Syndetic Set ◽  
Commuting Probability ◽  
Measure Preserving Actions

Let $T_{1}$ and $T_{2}$ be two commuting probability measure-preserving actions of a countable amenable group such that the group spanned by these actions acts ergodically. We show that ${\it\mu}(A\cap T_{1}^{g}A\cap T_{1}^{g}T_{2}^{g}A)>{\it\mu}(A)^{4}-{\it\epsilon}$ on a syndetic set for any measurable set $A$ and any ${\it\epsilon}>0$. The proof uses the concept of a sated system, introduced by Austin.

On a construction of unitary cocycles and the representation theory of amenable groups

1983 ◽  
Vol 3 (1) ◽  
pp. 129-133 ◽  
Author(s):  
Colin E. Sutherland
Keyword(s):  
Representation Theory ◽  
Probability Space ◽  
Amenable Group ◽  
Unitary Representations ◽  
Intertwining Operators ◽  
Type I ◽  
Amenable Groups

AbstractIf K is a countable amenable group acting freely and ergodically on a probability space (Γ, μ), and G is an arbitrary countable amenable group, we construct an injection of the space of unitary representations of G into the space of unitary 1-cocyles for K on (Γ, μ); this injection preserves intertwining operators. We apply this to show that for many of the standard non-type-I amenable groups H, the representation theory of H contains that of every countable amenable group.

Weak Convergence Is Not Strong Convergence For Amenable Groups

2001 ◽  
Vol 44 (2) ◽  
pp. 231-241 ◽  
Author(s):  
Joseph M. Rosenblatt ◽  
George A. Willis
Keyword(s):  
Weak Convergence ◽  
Strong Convergence ◽  
Cayley Graphs ◽  
Linear Equations ◽  
Amenable Group ◽  
Amenable Groups

AbstractLet G be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net (fα) of positive, normalized functions in L1(G) such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction.

Non-Bernoulli systems with completely positive entropy

2008 ◽  
Vol 28 (1) ◽  
pp. 87-124 ◽  
Author(s):  
A. H. DOOLEY ◽  
V. YA. GOLODETS ◽  
D. J. RUDOLPH ◽  
S. D. SINEL’SHCHIKOV
Keyword(s):  
Infinite Order ◽  
Amenable Group ◽  
Positive Entropy ◽  
Completely Positive ◽  
New Approach ◽  
Amenable Groups ◽  
Uncountable Family ◽  
Cutting And Stacking ◽  
Bernoulli Systems

AbstractA new approach to actions of countable amenable groups with completely positive entropy (cpe), allowing one to answer some basic questions in this field, was recently developed. The question of the existence of cpe actions which are not Bernoulli was raised. In this paper, we prove that every countable amenable groupG, which contains an element of infinite order, has non-Bernoulli cpe actions. In fact we can produce, for any$h \in (0, \infty ]$, an uncountable family of cpe actions of entropyh, which are pairwise automorphically non-isomorphic. These actions are given by a construction which we call co-induction. This construction is related to, but different from the standard induced action. We study the entropic properties of co-induction, proving that ifαGis co-induced from an actionαΓof a subgroup Γ, thenh(αG)=h(αΓ). We also prove that ifαΓis a non-Bernoulli cpe action of Γ, thenαGis also non-Bernoulli and cpe. Hence the problem of finding an uncountable family of pairwise non-isomorphic cpe actions of the same entropy is reduced to one of finding an uncountable family of non-Bernoulli cpe actions of$\mathbb Z$, which pairwise satisfy a property we call ‘uniform somewhat disjointness’. We construct such a family using refinements of the classical cutting and stacking methods.

scholarly journals Homological dimension of elementary amenable groups

2020 ◽  
Vol 2020 (766) ◽  
pp. 45-60
Author(s):  
Peter H. Kropholler ◽  
Conchita Martínez-Pérez
Keyword(s):  
Simple Group ◽  
Characteristic Zero ◽  
Amenable Group ◽  
Solvable Groups ◽  
Complete Information ◽  
Homological Dimension ◽  
Important Paper ◽  
Amenable Groups ◽  
Coefficient Ring ◽  
Special Case

AbstractIn this paper we prove that the homological dimension of an elementary amenable group over an arbitrary commutative coefficient ring is either infinite or equal to the Hirsch length of the group. Established theory gives simple group theoretical criteria for finiteness of homological dimension and so we can infer complete information about this invariant for elementary amenable groups. Stammbach proved the special case of solvable groups over coefficient fields of characteristic zero in an important paper dating from 1970.

scholarly journals On a Sumset Conjecture of Erdős

2015 ◽  
Vol 67 (4) ◽  
pp. 795-809 ◽  
Author(s):  
Mauro Di Nasso ◽  
Isaac Goldbring ◽  
Renling Jin ◽  
Steven Leth ◽  
Martino Lupini ◽  
...  
Keyword(s):  
Positive Solution ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Amenable Groups ◽  
Banach Density ◽  
Infinite Sets ◽  
Erdös Conjecture

AbstractErdős conjectured that for any set A ⊆ ℕ with positive lower asymptotic density, there are infinite sets B;C ⊆ ℕ such that B + C ⊆ A. We verify Erdős’ conjecture in the case where A has Banach density exceeding ½ . As a consequence, we prove that, for A ⊆ ℕ with positive Banach density (amuch weaker assumption than positive lower density), we can find infinite B;C ⊆ ℕ such that B+C is contained in the union of A and a translate of A. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erdős’ conjecture for subsets of the natural numbers that are pseudorandom.

scholarly journals Sofic entropy and amenable groups

2011 ◽  
Vol 32 (2) ◽  
pp. 427-466 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Group Action ◽  
Amenable Group ◽  
Group Actions ◽  
Amenable Groups ◽  
Orbit Equivalent ◽  
Sofic Entropy ◽  
Classical Entropy

AbstractIn previous work, the author introduced a measure-conjugacy invariant for sofic group actions called sofic entropy. Here, it is proven that the sofic entropy of an amenable group action equals its classical entropy. The proof uses a new measure-conjugacy invariant called upper-sofic entropy and a theorem of Rudolph and Weiss for the entropy of orbit-equivalent actions relative to the orbit changeσ-algebra.

Uniformly Continuous Functionals and M-Weakly Amenable Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1005-1019 ◽  
Author(s):  
Brian Forrest ◽  
Tianxuan Miao
Keyword(s):  
Compact Group ◽  
Amenable Group ◽  
Locally Compact Group ◽  
Approximate Identity ◽  
Fourier Algebra ◽  
Locally Compact ◽  
Amenable Groups ◽  
Invariant Means ◽  
Continuous Functionals ◽  
Uniformly Continuous

AbstractLet G be a locally compact group. Let AM(G) (A0(G))denote the closure of A(G), the Fourier algebra of G in the space of bounded (completely bounded) multipliers of A(G). We call a locally compact group M-weakly amenable if AM(G) has a bounded approximate identity. We will show that when G is M-weakly amenable, the algebras AM(G) and A0(G) have properties that are characteristic of the Fourier algebra of an amenable group. Along the way we show that the sets of topologically invariant means associated with these algebras have the same cardinality as those of the Fourier algebra.

Sign in / Sign up

Export Citation Format

Share Document