scholarly journals ELEMENTARY AMENABLE SUBGROUPS OF R. THOMPSON'S GROUP F

2005 ◽  
Vol 15 (04) ◽  
pp. 619-642 ◽  
Author(s):  
MATTHEW G. BRIN
Keyword(s):  
Piecewise Linear ◽  
Amenable Group ◽  
Solvable Groups ◽  
Amenable Groups ◽  
Elementary Class ◽  
Limit Ordinal ◽  
Thompson’S Group ◽  
Thompson's Group F

We study the subgroups of R. J. Thompson's group F and PLo(I), the group of orientation preserving, piecewise linear self homeomorphisms of [0, 1]. We exhibit, for each non-limit ordinal α ≤ ω2 + 1, an elementary amenable group of elementary class α (under Chou's stratification of elementary amenable groups) that is a subgroup of F and thus of PLo(I). We also give examples that negatively answer a question of Sapir about non-solvable groups in F and PLo(I).

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 growth of homology torsion in amenable groups

2016 ◽  
Vol 162 (2) ◽  
pp. 337-351 ◽  
Author(s):  
ADITI KAR ◽  
PETER KROPHOLLER ◽  
NIKOLAY NIKOLOV
Keyword(s):  
Simplicial Complex ◽  
Amenable Group ◽  
Solvable Groups ◽  
Integral Homology ◽  
Amenable Groups ◽  
Derived Length ◽  
Compact Quotient ◽  
Simply Connected

AbstractSuppose an amenable group G is acting freely on a simply connected simplicial complex $\~{X}$ with compact quotient X. Fix n ≥ 1, assume $H_n(\~{X}, \mathbb{Z}) = 0$ and let (Hi) be a Farber chain in G. We prove that the torsion of the integral homology in dimension n of $\~{X}/H_i$ grows subexponentially in [G : Hi]. This fails if X is not compact. We provide the first examples of amenable groups for which torsion in homology grows faster than any given function. These examples include some solvable groups of derived length 3 which is the minimal possible.

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.

Thompson’s Group F

2018 ◽  
pp. 37-66
Author(s):  
Marianna C. Bonanome ◽  
Margaret H. Dean ◽  
Judith Putnam Dean
Keyword(s):  
Thompson’S Group ◽  
Thompson's Group F

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 ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

2004 ◽  
Vol 14 (05n06) ◽  
pp. 677-702 ◽  
Author(s):  
V. S. GUBA
Keyword(s):  
Growth Rate ◽  
Lower Bound ◽  
Cayley Graph ◽  
Upper Bound ◽  
Growth Function ◽  
Vertex Degree ◽  
General Growth ◽  
Thompson’S Group ◽  
Average Vertex ◽  
Thompson's Group F

We study some properties of the Cayley graph of R. Thompson's group F in generators x0, x1. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x0, x1. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x0, x1 and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of [Formula: see text].

scholarly journals Random subgroups of Thompson’s group F

2010 ◽  
pp. 91-126 ◽  
Author(s):  
Sean Cleary ◽  
Murray Elder ◽  
Andrew Rechnitzer ◽  
Jennifer Taback
Keyword(s):  
Thompson’S Group ◽  
Thompson's Group F
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