Finitely Generated Amenable Groups

2010 ◽  
pp. 151-232
Author(s):  
Tullio Ceccherini-Silberstein ◽  
Michel Coornaert
Keyword(s):  
Finitely Generated ◽  
Amenable Groups

Infinitely presented permutation stable groups and invariant random subgroups of metabelian groups

2021 ◽  
pp. 1-36
Author(s):  
ARIE LEVIT ◽  
ALEXANDER LUBOTZKY
Keyword(s):  
Ergodic Theorem ◽  
Abelian Groups ◽  
Wreath Products ◽  
Finitely Generated ◽  
Metabelian Groups ◽  
Amenable Groups ◽  
Pointwise Ergodic Theorem ◽  
Lamplighter Group ◽  
Invariant Random Subgroups

Abstract We prove that all invariant random subgroups of the lamplighter group L are co-sofic. It follows that L is permutation stable, providing an example of an infinitely presented such group. Our proof applies more generally to all permutational wreath products of finitely generated abelian groups. We rely on the pointwise ergodic theorem for amenable groups.

scholarly journals On a theorem of Avez

2019 ◽  
Vol 22 (3) ◽  
pp. 383-395
Author(s):  
Murray Elder ◽  
Cameron Rogers
Keyword(s):  
Normal Subgroup ◽  
Probability Measure ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finitely Generated Group ◽  
Amenable Radical

Abstract For each symmetric, aperiodic probability measure μ on a finitely generated group G, we define a subset {A_{\mu}} consisting of group elements g for which the limit of the ratio {{\mu^{\ast n}(g)}/{\mu^{\ast n}(e)}} tends to 1. We prove that {A_{\mu}} is a subgroup, is amenable, contains every finite normal subgroup, and {G=A_{\mu}} if and only if G is amenable. For non-amenable groups we show that {A_{\mu}} is not always a normal subgroup and can depend on the measure. We formulate some conjectures relating {A_{\mu}} to the amenable radical.

scholarly journals Symbolic dynamics on amenable groups: the entropy of generic shifts

2016 ◽  
Vol 37 (4) ◽  
pp. 1187-1210 ◽  
Author(s):  
JOSHUA FRISCH ◽  
OMER TAMUZ
Keyword(s):  
Symbolic Dynamics ◽  
Amenable Group ◽  
Finite Alphabet ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Strongly Irreducible ◽  
Isolated Points ◽  
Zero Entropy ◽  
Weakly Mixing

Let$G$be a finitely generated amenable group. We study the space of shifts on$G$over a given finite alphabet $A$. We show that the zero entropy shifts are generic in this space, and that, more generally, the shifts of entropy$c$are generic in the space of shifts with entropy at least $c$. The same is shown to hold for the space of transitive shifts and for the space of weakly mixing shifts. As applications of this result, we show that, for every entropy value$c\in [0,\log |A|]$, there is a weakly mixing subshift of$A^{G}$with entropy $c$. We also show that the set of strongly irreducible shifts does not form a$G_{\unicode[STIX]{x1D6FF}}$in the space of shifts, and that all non-trivial, strongly irreducible shifts are non-isolated points in this space.

A NOTE ON COMMUTING SQUARES ARISING FROM AUTOMORPHISMS ON SUBFACTORS

1999 ◽  
Vol 10 (02) ◽  
pp. 207-214 ◽  
Author(s):  
PHAN H. LOI
Keyword(s):  
Finite Index ◽  
Simple Proof ◽  
Classification Theorem ◽  
Type Ii ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finite Set ◽  
Commuting Squares ◽  
Special Case

Using an idea due to Popa, we can associate a commuting square of factors to any given finite set of automorphisms acting on an inclusion of factors of finite index. We use this setting to obtain a simple proof of Popa's classification theorem of strongly outer actions of finitely generated discrete strongly amenable groups on a strongly amenable inclusion of type II 1 factors. We also obtain a new complete outer conjugacy invariant for arbitrary automorphisms, which contains the higher obstruction of Kawahigashi and the standard invariant as a special case.

scholarly journals Necessary conditions for tiling finitely generated amenable groups

2020 ◽  
Vol 40 (4) ◽  
pp. 2335-2346
Author(s):  
Benjamin Hellouin de Menibus ◽  
◽  
Hugo Maturana Cornejo ◽  
Keyword(s):  
Necessary Conditions ◽  
Finitely Generated ◽  
Amenable Groups

scholarly journals Computability of Følner sets

2017 ◽  
Vol 27 (07) ◽  
pp. 819-830 ◽  
Author(s):  
Matteo Cavaleri
Keyword(s):  
Word Problem ◽  
Finitely Generated ◽  
Amenable Groups ◽  
Finitely Generated Groups ◽  
Distortion Functions ◽  
Solvable Word Problem ◽  
Følner Sets ◽  
Finitely Presented ◽  
Unsolvable Word Problem ◽  
Solvable Word

We define the notion of computability of Følner sets for finitely generated amenable groups. We prove, by an explicit description, that the Kharlampovich groups, finitely presented solvable groups with unsolvable Word Problem, have computable Følner sets. We also prove computability of Følner sets for extensions — with subrecursive distortion functions — of amenable groups with solvable Word Problem by finitely generated groups with computable Følner sets. Moreover, we obtain some known and some new upper bounds for the Følner function for these particular extensions.

scholarly journals Non-elementary amenable subgroups of automata groups

2017 ◽  
Vol 10 (01) ◽  
pp. 35-45
Author(s):  
Kate Juschenko
Keyword(s):  
Finitely Generated ◽  
Locally Finite ◽  
Amenable Groups ◽  
Automata Groups ◽  
Groups Of Automorphisms

We consider groups of automorphisms of rooted locally finite trees, and give conditions on its subgroups that imply that they are not elementary amenable. We give a unified proof for all known examples of non-elementary amenable groups that act on the trees: groups of intermediate growths and Basilica group. Moreover, we show that all finitely generated branch groups are not elementary amenable, which was conjectured by Grigorchuk.

Automorphisms of finite order of nilpotent groups II

2014 ◽  
Vol 51 (4) ◽  
pp. 547-555 ◽  
Author(s):  
B. Wehrfritz
Keyword(s):  
Nilpotent Group ◽  
Finite Order ◽  
Finite Index ◽  
Nilpotent Groups ◽  
Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

Collectives of Automata in Finitely Generated Groups

2020 ◽  
Vol 108 (5-6) ◽  
pp. 671-678
Author(s):  
D. V. Gusev ◽  
I. A. Ivanov-Pogodaev ◽  
A. Ya. Kanel-Belov
Keyword(s):  
Finitely Generated ◽  
Finitely Generated Groups
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