On graph products of multipliers and the Haagerup property for -dynamical systems

2019 ◽  
Vol 40 (12) ◽  
pp. 3188-3216
Author(s):  
SCOTT ATKINSON
Keyword(s):  
Dynamical Systems ◽  
Positive Definite ◽  
Alternative Proof ◽  
Discrete Groups ◽  
Graph Products ◽  
Graph Product ◽  
Haagerup Property ◽  
Products Of Groups ◽  
Product Action

We consider the notion of the graph product of actions of discrete groups $\{G_{v}\}$ on a $C^{\ast }$-algebra ${\mathcal{A}}$ and show that under suitable commutativity conditions the graph product action $\star _{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D6FC}_{v}:\star _{\unicode[STIX]{x1D6E4}}G_{v}\curvearrowright {\mathcal{A}}$ has the Haagerup property if each action $\unicode[STIX]{x1D6FC}_{v}:G_{v}\curvearrowright {\mathcal{A}}$ possesses the Haagerup property. This generalizes the known results on graph products of groups with the Haagerup property. To accomplish this, we introduce the graph product of multipliers associated to the actions and show that the graph product of positive-definite multipliers is positive definite. These results have impacts on left-transformation groupoids and give an alternative proof of a known result for coarse embeddability. We also record a cohomological characterization of the Haagerup property for group actions.

scholarly journals GRAPH PRODUCTS OF RIGHT CANCELLATIVE MONOIDS

2009 ◽  
Vol 87 (2) ◽  
pp. 227-252 ◽  
Author(s):  
JOHN FOUNTAIN ◽  
MARK KAMBITES
Keyword(s):  
Graph Products ◽  
Graph Product ◽  
General Characterization ◽  
Inverse Hull ◽  
Cancellative Monoids

AbstractOur first main result shows that a graph product of right cancellative monoids is itself right cancellative. If each of the component monoids satisfies the condition that the intersection of two principal left ideals is either principal or empty, then so does the graph product. Our second main result gives a presentation for the inverse hull of such a graph product. We then specialize to the case of the inverse hulls of graph monoids, obtaining what we call ‘polygraph monoids’. Among other properties, we observe that polygraph monoids are F*-inverse. This follows from a general characterization of those right cancellative monoids with inverse hulls that are F*-inverse.

ORDERING GRAPH PRODUCTS OF GROUPS

2012 ◽  
Vol 22 (04) ◽  
pp. 1250037 ◽  
Author(s):  
I. M. CHISWELL
Keyword(s):  
Graph Products ◽  
Graph Product ◽  
Free Products ◽  
Products Of Groups

It is shown that a graph product of right-orderable groups is right orderable, and that a graph product of (two-sided) orderable groups is orderable. The latter result makes use of a new way of ordering free products of groups.

scholarly journals Separability properties of automorphisms of graph products of groups

2016 ◽  
Vol 26 (01) ◽  
pp. 1-27
Author(s):  
Michal Ferov
Keyword(s):  
Finite Groups ◽  
Nilpotent Groups ◽  
Technical Condition ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Residually Finite ◽  
Products Of Groups ◽  
Text Version ◽  
Inner Automorphisms

We study properties of automorphisms of graph products of groups. We show that graph product [Formula: see text] has nontrivial pointwise inner automorphisms if and only if some vertex group corresponding to a central vertex has nontrivial pointwise inner automorphisms. We use this result to study residual finiteness of [Formula: see text]. We show that if all vertex groups are finitely generated residually finite and the vertex groups corresponding to central vertices satisfy certain technical (yet natural) condition, then [Formula: see text] is residually finite. Finally, we generalize this result to graph products of residually [Formula: see text]-finite groups to show that if [Formula: see text] is a graph product of finitely generated residually [Formula: see text]-finite groups such that the vertex groups corresponding to central vertices satisfy the [Formula: see text]-version of the technical condition then [Formula: see text] is virtually residually [Formula: see text]-finite. We use this result to prove bi-orderability of Torreli groups of some graph products of finitely generated residually torsion-free nilpotent groups.

scholarly journals SURFACE SUBGROUPS OF GRAPH PRODUCTS OF GROUPS

2012 ◽  
Vol 22 (08) ◽  
pp. 1240003
Author(s):  
SANG-HYUN KIM
Keyword(s):  
Direct Product ◽  
Finite Groups ◽  
Coxeter Groups ◽  
Surface Group ◽  
Hyperbolic Surface ◽  
Graph Products ◽  
Graph Product ◽  
Cyclic Groups ◽  
Products Of Groups ◽  
Product Kernel

Let G be a graph product of a collection of groups and H be the direct product of the same collection of groups, so that there is a natural surjection p : G → H. The kernel of this map p is called a graph product kernel. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.

scholarly journals The topology of graph products of groups

1994 ◽  
Vol 37 (3) ◽  
pp. 539-544
Author(s):  
John Meier
Keyword(s):  
Isoperimetric Inequality ◽  
Free Product ◽  
Graph Products ◽  
Graph Product ◽  
Infinite Groups ◽  
Products Of Groups ◽  
Explicit Bounds ◽  
Finitely Presented

Given a finite (connected) simplicial graph with groups assigned to the vertices, the graph product of the vertex groups is the free product modulo the relation that adjacent groups commute. The graph product of finitely presented infinite groups is both semistable at infinity and quasi-simply filtrated. Explicit bounds for the isoperimetric inequality and isodiametric inequality for graph products is given, based on isoperimetric and isodiametric inequalities for the vertex groups.

scholarly journals Tits alternatives for graph products

2015 ◽  
Vol 2015 (704) ◽  
Author(s):  
Yago Antolín ◽  
Ashot Minasyan
Keyword(s):  
Graph Products ◽  
Parabolic Subgroups ◽  
Graph Product ◽  
Natural Conditions ◽  
Tits Alternative ◽  
Products Of Groups ◽  
Rank 2 ◽  
The Stability ◽  
Free Subgroups ◽  
Product Of Groups

AbstractWe discuss various types of Tits alternative for subgroups of graph products of groups, and prove that, under some natural conditions, a graph product of groups satisfies a given form of Tits alternative if and only if each vertex group satisfies this alternative. As a corollary, we show that every finitely generated subgroup of a graph product of virtually solvable groups is either virtually solvable or large. As another corollary, we prove that every non-abelian subgroup of a right angled Artin group has an epimorphism onto the free group of rank 2. In the course of the paper we develop the theory of parabolic subgroups, which allows to describe the structure of subgroups of graph products that contain no non-abelian free subgroups. We also obtain a number of results regarding the stability of some group properties under taking graph products.

scholarly journals On equicontinuous factors of flows on locally path-connected compact spaces

2020 ◽  
pp. 1-18
Author(s):  
NIKOLAI EDEKO
Keyword(s):  
Lie Group ◽  
Metric Space ◽  
Betti Number ◽  
Alternative Proof ◽  
Simple Consequence ◽  
Compact Metric Space ◽  
Invariant Functions ◽  
Compact Spaces ◽  
Path Connected

Abstract We consider a locally path-connected compact metric space K with finite first Betti number $\textrm {b}_1(K)$ and a flow $(K, G)$ on K such that G is abelian and all G-invariant functions $f\,{\in}\, \text{\rm C}(K)$ are constant. We prove that every equicontinuous factor of the flow $(K, G)$ is isomorphic to a flow on a compact abelian Lie group of dimension less than ${\textrm {b}_1(K)}/{\textrm {b}_0(K)}$ . For this purpose, we use and provide a new proof for Theorem 2.12 of Hauser and Jäger [Monotonicity of maximal equicontinuous factors and an application to toral flows. Proc. Amer. Math. Soc.147 (2019), 4539–4554], which states that for a flow on a locally connected compact space the quotient map onto the maximal equicontinuous factor is monotone, i.e., has connected fibers. Our alternative proof is a simple consequence of a new characterization of the monotonicity of a quotient map $p\colon K\to L$ between locally connected compact spaces K and L that we obtain by characterizing the local connectedness of K in terms of the Banach lattice $\textrm {C}(K)$ .

A UNIFIED FRAMEWORK FOR SYNCHRONIZATION AND CONTROL OF DYNAMICAL SYSTEMS

1994 ◽  
Vol 04 (04) ◽  
pp. 979-998 ◽  
Author(s):  
CHAI WAH WU ◽  
LEON O. CHUA
Keyword(s):  
Dynamical Systems ◽  
Chaotic Systems ◽  
Main Tool ◽  
Lyapunov Stability Theory ◽  
Asymptotical Stability ◽  
Unified Framework ◽  
Partial Synchronization ◽  
Chua’S Oscillator ◽  
And Control

In this paper, we give a framework for synchronization of dynamical systems which unifies many results in synchronization and control of dynamical systems, in particular chaotic systems. We define concepts such as asymptotical synchronization, partial synchronization and synchronization error bounds. We show how asymptotical synchronization is related to asymptotical stability. The main tool we use to prove asymptotical stability and synchronization is Lyapunov stability theory. We illustrate how many previous results on synchronization and control of chaotic systems can be derived from this framework. We will also give a characterization of robustness of synchronization and show that master-slave asymptotical synchronization in Chua’s oscillator is robust.

Numerical characterization of nonlinear dynamical systems using parallel computing: The role of GPUs approach

2016 ◽  
Vol 37 ◽  
pp. 143-162 ◽  
Author(s):  
Filipe I. Fazanaro ◽  
Diogo C. Soriano ◽  
Ricardo Suyama ◽  
Marconi K. Madrid ◽  
José Raimundo de Oliveira ◽  
...  
Keyword(s):  
Parallel Computing ◽  
Dynamical Systems ◽  
Nonlinear Dynamical Systems ◽  
Nonlinear Dynamical ◽  
Numerical Characterization

Canonical rules

2009 ◽  
Vol 74 (4) ◽  
pp. 1171-1205 ◽  
Author(s):  
Emil Jeřábek
Keyword(s):  
Alternative Proof ◽  
Finite Model ◽  
Consequence Relations ◽  
Model Property ◽  
Finite Model Property ◽  
Admissible Rules ◽  
Superintuitionistic Logics ◽  
Canonical Formulas ◽  
Global Consequence

AbstractWe develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

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