scholarly journals Continuous Cocycle Superrigidity for the Full Shift Over a Finitely Generated Torsion Group

2018 ◽  
Vol 2020 (6) ◽  
pp. 1610-1620 ◽  
Author(s):  
David Bruce Cohen
Keyword(s):  
Discrete Group ◽  
Infinite Order ◽  
Torsion Group ◽  
Finitely Generated ◽  
Full Shift ◽  
Order Element

Abstract Chung and Jiang showed that if a one-ended group contains an infinite order element, then every continuous cocycle over the full shift, taking values in a discrete group, must be cohomologous to a homomorphism. We show that their conclusion holds for all one-ended groups, so that the hypothesis of admitting an infinite order element may be omitted.

Convex fundamental domains of Fuchsian groups

1974 ◽  
Vol 76 (3) ◽  
pp. 511-513 ◽  
Author(s):  
A. F. Beardon
Keyword(s):  
Complex Plane ◽  
Unit Disc ◽  
Fuchsian Group ◽  
Discrete Group ◽  
Fuchsian Groups ◽  
Finitely Generated ◽  
Möbius Transformations ◽  
Fundamental Domains ◽  
Mobius Transformations

In this paper a Fuchsian group G shall be a discrete group of Möbius transformations each of which maps the unit disc △ in the complex plane onto itself. We shall also assume throughout this paper that G is both finitely generated and of the first kind.

scholarly journals APPROXIMATING THE FIRST L2-BETTI NUMBER OF RESIDUALLY FINITE GROUPS

2011 ◽  
Vol 03 (02) ◽  
pp. 153-160 ◽  
Author(s):  
W. LÜCK ◽  
D. OSIN
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Betti Number ◽  
Betti Numbers ◽  
Torsion Group ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups

We show that the first L2-betti number of a finitely generated residually finite group can be estimated from below by using ordinary first betti numbers of finite index normal subgroups. As an application, we construct a finitely generated infinite residually finite torsion group with positive first L2-betti number.

On theorems of Thornton

1972 ◽  
Vol 6 (2) ◽  
pp. 211-212 ◽  
Author(s):  
R. Lalithambal
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Discrete Group ◽  
Torsion Group ◽  
Additional Hypothesis

The topology of a topological group in which the intersection of open sets is open is uniquely determined by a normal subgroup, and the group is uniquely an extension of an indiscrete group by a discrete group. This was proved by M.C. Thornton under the additional hypothesis that the group is a torsion group. The proofs here given make the more general facts almost trivial.

Finitely approximate groups and actions Part I: The Ribes–Zalesskiĭ property

2011 ◽  
Vol 76 (4) ◽  
pp. 1297-1306 ◽  
Author(s):  
Christian Rosendal
Keyword(s):  
Metric Space ◽  
Discrete Group ◽  
Metric Spaces ◽  
Finitely Generated ◽  
Partial Isometries ◽  
Profinite Topology

AbstractWe investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.

Locally finite groups and configurations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Mahdi Meisami ◽  
Ali Rejali ◽  
Meisam Soleimani Malekan ◽  
Akram Yousofzadeh
Keyword(s):  
Finite Group ◽  
Positive Solution ◽  
Finite Groups ◽  
Discrete Group ◽  
Locally Finite Group ◽  
Finitely Generated ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Normalized Solution

Abstract Let 𝐺 be a discrete group. In 2001, Rosenblatt and Willis proved that 𝐺 is amenable if and only if every possible system of configuration equations admits a normalized solution. In this paper, we show independently that 𝐺 is locally finite if and only if every possible system of configuration equations admits a strictly positive solution. Also, we give a procedure to get equidecomposable subsets 𝐴 and 𝐵 of an infinite finitely generated or a locally finite group 𝐺 such that A ⊊ B A\subsetneq B , directly from a system of configuration equations not having a strictly positive solution.

On autocommutators and the autocommutator subgroup in infinite abelian groups

2018 ◽  
Vol 30 (4) ◽  
pp. 877-885
Author(s):  
Luise-Charlotte Kappe ◽  
Patrizia Longobardi ◽  
Mercede Maj
Keyword(s):  
Automorphism Group ◽  
Abelian Groups ◽  
Torsion Group ◽  
Nilpotency Class ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Minimal Order ◽  
Autocommutator Subgroup ◽  
Free Abelian Groups ◽  
Torsion Free

Abstract It is well known that the set of commutators in a group usually does not form a subgroup. A similar phenomenon occurs for the set of autocommutators. There exists a group of order 64 and nilpotency class 2, where the set of autocommutators does not form a subgroup, and this group is of minimal order with this property. However, for finite abelian groups, the set of autocommutators is always a subgroup. We will show in this paper that this is no longer true for infinite abelian groups. We characterize finitely generated infinite abelian groups in which the set of autocommutators does not form a subgroup and show that in an infinite abelian torsion group the set of commutators is a subgroup. Lastly, we investigate torsion-free abelian groups with finite automorphism group and we study whether the set of autocommutators forms a subgroup in those groups.

scholarly journals ON THE CAPABILITY OF FINITELY GENERATED NON-TORSION GROUPS OF NILPOTENCY CLASS 2

2011 ◽  
Vol 53 (2) ◽  
pp. 411-417 ◽  
Author(s):  
LUISE-CHARLOTTE KAPPE ◽  
NOR MUHAINIAH MOHD ALI ◽  
NOR HANIZA SARMIN
Keyword(s):  
Torsion Group ◽  
Nilpotency Class ◽  
Necessary Condition ◽  
Finitely Generated ◽  
Free Rank ◽  
Torsion Groups ◽  
Necessary And Sufficient ◽  
Torsion Free Rank ◽  
Torsion Free

AbstractA group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group.

scholarly journals Spectral gap property and strong ergodicity for groups of affine transformations of solenoids

2018 ◽  
Vol 40 (5) ◽  
pp. 1180-1193
Author(s):  
BACHIR BEKKA ◽  
CAMILLE FRANCINI
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Solvable Group ◽  
Haar Measure ◽  
Discrete Group ◽  
Spectral Gap ◽  
Canonical Projection ◽  
Affine Transformations ◽  
Finitely Generated ◽  
Finite Dimensional

Let $X$ be a solenoid, i.e. a compact, finite-dimensional, connected abelian group with normalized Haar measure $\unicode[STIX]{x1D707}$, and let $\unicode[STIX]{x1D6E4}\rightarrow \operatorname{Aff}(X)$ be an action of a countable discrete group $\unicode[STIX]{x1D6E4}$ by continuous affine transformations of $X$. We show that the probability measure preserving action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ does not have the spectral gap property if and only if there exists a $p_{\text{a}}(\unicode[STIX]{x1D6E4})$-invariant proper subsolenoid $Y$ of $X$ such that the image of $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually solvable group, where $p_{\text{a}}:\operatorname{Aff}(X)\rightarrow \operatorname{Aut}(X)$ is the canonical projection. When $\unicode[STIX]{x1D6E4}$ is finitely generated or when $X$ is the $a$-adic solenoid for an integer $a\geq 1$, the subsolenoid $Y$ can be chosen so that the image $\unicode[STIX]{x1D6E4}$ in $\operatorname{Aff}(X/Y)$ is a virtually abelian group. In particular, an action $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ by affine transformations on a solenoid $X$ has the spectral gap property if and only if $\unicode[STIX]{x1D6E4}\curvearrowright (X,\unicode[STIX]{x1D707})$ is strongly ergodic.

Comments On a Discreteness Condition for Subgroups of SL(2, C)

1979 ◽  
Vol 31 (1) ◽  
pp. 87-92 ◽  
Author(s):  
Troels Jørgensen
Keyword(s):  
Discrete Group ◽  
Infinite Order ◽  
Convergent Sequence ◽  
Image Position ◽  
Discreteness Condition

SL(2,C) is the group of all complex unimodular 2 × 2 matrices. A subgroup of SL(2, C) is said to be discrete if it does not contain any convergent sequence of distinct elements. A subgroup is said to be elementary if the commutator of any two elements of infinite order has trace 2. The discreteness condition which this note relates to is the following:PROPOSITION 1. If two complex, unimodular 2 × 2 matrices X and Y generatea non-elementary, discrete group, then

THE BURNSIDE PROBLEM FOR GROUPS OF LOW QUADRATIC GROWTH

1996 ◽  
Vol 06 (03) ◽  
pp. 369-377
Author(s):  
ROBERTO INCITTI
Keyword(s):  
Infinite Order ◽  
Polynomial Growth ◽  
Growth Function ◽  
Infinite Group ◽  
Quadratic Growth ◽  
Burnside Problem ◽  
Finitely Generated ◽  
Combinatorial Argument ◽  
Groups Of Polynomial Growth ◽  
Generating System

We show with a combinatorial argument that a finitely generated infinite group whose growth function relative to some finite generating system is less or equal to [Formula: see text], r<2, contains an element of infinite order. This result is aimed at investigating the combinatorial nature of M. Gromov’s theorem on groups of polynomial growth.

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