scholarly journals The Conjugacy Ratio of Groups

2019 ◽  
Vol 62 (3) ◽  
pp. 895-911 ◽  
Author(s):  
Laura Ciobanu ◽  
Charles Garnet Cox ◽  
Armando Martino
Keyword(s):  
Finite Groups ◽  
Hyperbolic Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Subexponential Growth ◽  
Finitely Generated Group ◽  
Growth Functions ◽  
Lamplighter Group ◽  
Residually Finite Groups ◽  
Right Angled Artin Groups

AbstractIn this paper we introduce and study the conjugacy ratio of a finitely generated group, which is the limit at infinity of the quotient of the conjugacy and standard growth functions. We conjecture that the conjugacy ratio is 0 for all groups except the virtually abelian ones, and confirm this conjecture for certain residually finite groups of subexponential growth, hyperbolic groups, right-angled Artin groups and the lamplighter group.

scholarly journals An uncountable family of finitely generated residually finite groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Hip Kuen Chong ◽  
Daniel T. Wise
Keyword(s):  
Free Group ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Uncountable Family ◽  
Residually Finite Groups ◽  
Rank 2

Abstract We study a family of finitely generated residually finite groups. These groups are doubles F 2 * H F 2 F_{2}*_{H}F_{2} of a rank-2 free group F 2 F_{2} along an infinitely generated subgroup 𝐻. Varying 𝐻 yields uncountably many groups up to isomorphism.

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.

POLYNOMIAL INDEX GROWTH GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 773-782 ◽  
Author(s):  
ANTAL BALOG ◽  
LÁSZLÓ PYBER ◽  
AVINOAM MANN
Keyword(s):  
Finite Groups ◽  
Simple Groups ◽  
Finitely Generated ◽  
Constant Power ◽  
Residually Finite ◽  
Almost Simple Groups ◽  
Small Constant ◽  
Residually Finite Groups ◽  
Strengthened Form

We show that the order of a finite simple of Lie type is bounded by a small constant power of its exponent. This confirms, in a strengthened form, a conjecture of Vaughan-Lee and Zel'manov on the order and exponent of almost simple groups. We also obtain various structural restrictions on groups of polynomial index growth. Combining the above results we construct finitely generated residually finite groups of polynomial index growth which are neither linear nor boundedly generated. This answers questions of Segal and Platonov–Rapinchuk respectively. A further question of Platonov–Rapinchuk concerning a weakened polynomial index growth assumption is also answered.

scholarly journals Some properties of endomorphisms in residually finite groups

1977 ◽  
Vol 24 (1) ◽  
pp. 117-120 ◽  
Author(s):  
Ronald Hirshon
Keyword(s):  
Finite Group ◽  
Positive Integer ◽  
Free Product ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Residually Finite Groups ◽  
A Finite Group ◽  
Torsion Free

AbstractIf ε is an endomorphism of a finitely generated residually finite group onto a subgroup Fε of finite index in F, then there exists a positive integer k such that ε is an isomorphism of Fεk. If K is the kernel of ε, then K is a finite group so that if F is a non trivial free product or if F is torsion free, then ε is an isomorphism on F. If ε is an endomorphism of a finitely generated resedually finite group onto a subgroup Fε (not necessatily of ginite index in F) and if the kernel of ε obeys the minimal condition for subgroups then there exists a positive integer k such that ε is an isomorphism on Fεk.

scholarly journals On the residual and profinite closures of commensurated subgroups

2019 ◽  
Vol 169 (2) ◽  
pp. 411-432
Author(s):  
PIERRE–EMMANUEL CAPRACE ◽  
PETER H. KROPHOLLER ◽  
COLIN D. REID ◽  
PHILLIP WESOLEK
Keyword(s):  
Finite Groups ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Infinite Groups ◽  
Residually Finite ◽  
Finitely Generated Groups ◽  
Residually Finite Groups ◽  
Groups Acting On Trees ◽  
Polycyclic Groups

AbstractThe residual closure of a subgroup H of a group G is the intersection of all virtually normal subgroups of G containing H. We show that if G is generated by finitely many cosets of H and if H is commensurated, then the residual closure of H in G is virtually normal. This implies that separable commensurated subgroups of finitely generated groups are virtually normal. A stream of applications to separable subgroups, polycyclic groups, residually finite groups, groups acting on trees, lattices in products of trees and just-infinite groups then flows from this main result.

scholarly journals Detecting ends of residually finite groups in profinite completions

2013 ◽  
Vol 155 (3) ◽  
pp. 379-389 ◽  
Author(s):  
OWEN COTTON–BARRATT
Keyword(s):  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Finitely Generated Groups ◽  
Residually Finite Groups ◽  
Number Of Ends ◽  
Profinite Completions ◽  
Property T

AbstractLet $\mathcal{C}$ be a variety of finite groups. We use profinite Bass--Serre theory to show that if u : H ↪ G is a map of finitely generated residually $\mathcal{C}$ groups such that the induced map û : Ĥ → Ĝ is a surjection of the pro-$\mathcal{C}$ completions, and G has more than one end, then H has the same number of ends as G. However if G has one end the number of ends of H may be larger; we observe cases where this occurs for $\mathcal{C}$ the class of finite p-groups.We produce a monomorphism of groups u : H ↪ G such that: either G is hyperbolic but not residually finite; or û : Ĥ → Ĝ is an isomorphism of profinite completions but H has property (T) (and hence (FA)), but G has neither. Either possibility would give new examples of pathological finitely generated groups.

scholarly journals Rokhlin dimension for actions of residually finite groups

2017 ◽  
Vol 39 (8) ◽  
pp. 2248-2304 ◽  
Author(s):  
GÁBOR SZABÓ ◽  
JIANCHAO WU ◽  
JOACHIM ZACHARIAS
Keyword(s):  
Large Class ◽  
Finite Groups ◽  
Metric Spaces ◽  
Nilpotent Groups ◽  
Asymptotic Dimension ◽  
Finitely Generated ◽  
Residually Finite ◽  
Compact Metric Spaces ◽  
Finite Dimensional ◽  
Residually Finite Groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on $\text{C}^{\ast }$-algebras, extending previous such notions for actions of finite groups and the integers by Hirshberg, Winter and the third author. We are able to extend most of their results to a much larger class of groups: those admitting box spaces of finite asymptotic dimension. This latter condition is a refinement of finite asymptotic dimension and has not previously been considered. In a detailed study we show that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a large class of examples. We show that actions with finite Rokhlin dimension by groups with finite-dimensional box spaces preserve the property of having finite nuclear dimension when passing to the crossed product $\text{C}^{\ast }$-algebra. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite-dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group$\text{C}^{\ast }$-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^{m}$-actions by the first author to a significantly larger class of groups, showing that a large class of crossed products by actions of such groups fall under the remit of the Elliott classification programme. We also provide results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing $\text{C}^{\ast }$-algebra.

A property of finitely generated residually finite groups

1976 ◽  
Vol 15 (3) ◽  
pp. 347-350 ◽  
Author(s):  
P.F. Pickel
Keyword(s):  
Finite Group ◽  
Homomorphic Image ◽  
Finite Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Residually Finite Group ◽  
Isomorphism Classes ◽  
Residually Finite Groups ◽  
Finite Quotients ◽  
Relatively Free Group

Let F(G) denote the set of isomorphism classes of finite quotients of the group G. We say that groups G and H have isomorphic finite quotients (IFQ) if F(G) = F(H). In this note, we show that a finitely generated residually finite group G cannot have the same finite quotients as a proper homomorphic image (G is IFQ hopfian). We then obtain some results on groups with the same finite quotients as a relatively free group.

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