scholarly journals Complexity among the finitely generated subgroups of Thompson's group

2021 ◽  
Vol 5 (1) ◽  
pp. 1-58
Author(s):  
Collin Bleak ◽  
Matthew Brin ◽  
Justin Tatch Moore
Keyword(s):  
Finitely Generated ◽  
Thompson’S Group

scholarly journals Burghelea conjecture and asymptotic dimension of groups

2018 ◽  
Vol 12 (02) ◽  
pp. 321-356
Author(s):  
Alexander Engel ◽  
Michał Marcinkowski
Keyword(s):  
Cyclic Homology ◽  
Asymptotic Dimension ◽  
Group Rings ◽  
Finitely Generated ◽  
Type Formula ◽  
Complex Group ◽  
Bass Conjecture ◽  
Thompson’S Group ◽  
Decomposition Complexity

We review the Burghelea conjecture, which constitutes a full computation of the periodic cyclic homology of complex group rings, and its relation to the algebraic Baum–Connes conjecture. The Burghelea conjecture implies the Bass conjecture. We state two conjectures about groups of finite asymptotic dimension, which together imply the Burghelea conjecture for such groups. We prove both conjectures for many classes of groups. It is known that the Burghelea conjecture does not hold for all groups, although no finitely presentable counterexample was known. We construct a finitely presentable (even type [Formula: see text]) counterexample based on Thompson’s group [Formula: see text]. We construct as well a finitely generated counterexample with finite decomposition complexity.

scholarly journals Non-triviality of the Poisson boundary of random walks on the group of Monod

2019 ◽  
pp. 1-30
Author(s):  
BOGDAN STANKOV
Keyword(s):  
Random Walks ◽  
Sufficient Conditions ◽  
London Mathematical Society ◽  
Lecture Note ◽  
Poisson Boundary ◽  
Finitely Generated ◽  
Cambridge University ◽  
Lecture Note Series ◽  
Thompson’S Group ◽  
First Moment

We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on $H(\mathbb{Z})$ and its subgroups. The group $H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup $H$ of $H(\mathbb{Z})$ , we prove that either $H$ is solvable or every measure on $H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group $F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group $F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].

The BNSR-invariants of the Stein group 𝐹2,3

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Robert Spahn ◽  
Matthew C. B. Zaremsky
Keyword(s):  
Normal Subgroup ◽  
Piecewise Linear ◽  
Unit Interval ◽  
Finitely Generated ◽  
Discrete Character ◽  
Sigma 1 ◽  
Thompson’S Group ◽  
Finitely Presented

Abstract The Stein group F 2 , 3 F_{2,3} is the group of orientation-preserving piecewise linear homeomorphisms of the unit interval with slopes of the form 2 p ⁢ 3 q 2^{p}3^{q} ( p , q ∈ Z p,q\in\mathbb{Z} ) and breakpoints in Z ⁢ [ 1 6 ] \mathbb{Z}[\frac{1}{6}] . This is a natural relative of Thompson’s group 𝐹. In this paper, we compute the Bieri–Neumann–Strebel–Renz (BNSR) invariants Σ m ⁢ ( F 2 , 3 ) \Sigma^{m}(F_{2,3}) of the Stein group for all m ∈ N m\in\mathbb{N} . A consequence of our computation is that (as with 𝐹) every finitely presented normal subgroup of F 2 , 3 F_{2,3} is of type F ∞ \operatorname{F}_{\infty} . Another, more surprising, consequence is that (unlike 𝐹) the kernel of any map F 2 , 3 → Z F_{2,3}\to\mathbb{Z} is of type F ∞ \operatorname{F}_{\infty} , even though there exist maps F 2 , 3 → Z 2 F_{2,3}\to\mathbb{Z}^{2} whose kernels are not even finitely generated. In terms of BNSR-invariants, this means that every discrete character lies in Σ ∞ ⁢ ( F 2 , 3 ) \Sigma^{\infty}(F_{2,3}) , but there exist (non-discrete) characters that do not even lie in Σ 1 ⁢ ( F 2 , 3 ) \Sigma^{1}(F_{2,3}) . To the best of our knowledge, F 2 , 3 F_{2,3} is the first group whose BNSR-invariants are known exhibiting these properties.

scholarly journals A dynamical definition of f.g. virtually free groups

2016 ◽  
Vol 26 (01) ◽  
pp. 105-121 ◽  
Author(s):  
Daniel Bennett ◽  
Collin Bleak
Keyword(s):  
Word Problem ◽  
Formal Language ◽  
Dynamical Behavior ◽  
Free Groups ◽  
Finitely Generated ◽  
Closure Properties ◽  
Thompson’S Group ◽  
Definition Of ◽  
Context Free ◽  
Free Word

We show that the class of finitely generated virtually free groups is precisely the class of finitely generated demonstrable subgroups for Thompson’s group [Formula: see text]. The class of demonstrable groups for [Formula: see text] consists of all groups which can embed into [Formula: see text] with a natural dynamical behavior in their induced actions on the Cantor space [Formula: see text]. There are also connections with formal language theory, as the class of groups with context-free word problem is also the class of finitely generated virtually free groups, while Thompson’s group [Formula: see text] is a candidate as a universal [Formula: see text] group by Lehnert’s conjecture, corresponding to the class of groups with context free co-word problem (as introduced by Holt, Rees, Röver, and Thomas). Our main results answers a question of Berns-Zieve, Fry, Gillings, Hoganson, and Matthews, and separately of Bleak and Salazar-Díaz, and it fits into the larger exploration of the class of [Formula: see text] groups as it shows that all four of the known closure properties of the class of [Formula: see text] groups hold for the set of finitely generated subgroups of [Formula: see text].

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

scholarly journals TEST IDEALS IN RINGS WITH FINITELY GENERATED ANTI-CANONICAL ALGEBRAS – CORRIGENDUM

2016 ◽  
Vol 17 (4) ◽  
pp. 979-980
Author(s):  
Alberto Chiecchio ◽  
Florian Enescu ◽  
Lance Edward Miller ◽  
Karl Schwede
Keyword(s):  
Finitely Generated

Projectivity of finitely generated flat modules over semilocal rings

1985 ◽  
Vol 37 (2) ◽  
pp. 85-90
Author(s):  
I. I. Sakhaev
Keyword(s):  
Finitely Generated ◽  
Flat Modules

Primitivity in representations of polycyclic groups

1980 ◽  
Vol 88 (1) ◽  
pp. 15-31 ◽  
Author(s):  
D. L. Harper
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Nilpotent Group ◽  
Infinite Dimension ◽  
Finitely Generated ◽  
Polycyclic Groups

In an earlier paper (5) we showed that a finitely generated nilpotent group which is not abelian-by-finite has a primitive irreducible representation of infinite dimension over any non-absolute field. Here we are concerned primarily with the converse question: Suppose that G is a polycyclic-by-finite group with such a representation, then what can be said about G?

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