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.

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.

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 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.

WORD EQUATIONS OVER GRAPH PRODUCTS

2008 ◽  
Vol 18 (03) ◽  
pp. 493-533 ◽  
Author(s):  
VOLKER DIEKERT ◽  
MARKUS LOHREY
Keyword(s):  
Graph Products ◽  
Graph Product ◽  
Positive Theory ◽  
Cancellation Condition ◽  
Existential Theory ◽  
Word Equations ◽  
Product Of Groups

For monoids that satisfy a weak cancellation condition, it is shown that the decidability of the existential theory of word equations is preserved under graph products. Furthermore, it is shown that the positive theory of a graph product of groups can be reduced to the positive theories of those factors, which commute with all other factors, and the existential theories of the remaining factors. Both results also include suitable constraints for the variables. Larger classes of constraints lead in many cases to undecidability results.

scholarly journals RIGIDITY OF GRAPH PRODUCTS OF ABELIAN GROUPS

2008 ◽  
Vol 77 (2) ◽  
pp. 187-196 ◽  
Author(s):  
MAURICIO GUTIERREZ ◽  
ADAM PIGGOTT
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Abelian Groups ◽  
Graph Products ◽  
Vertex Group ◽  
Graph Product ◽  
Finitely Generated ◽  
Product Decomposition ◽  
Unique Decomposition ◽  
Product Of Groups

AbstractWe show that if G is a group and G has a graph-product decomposition with finitely generated abelian vertex groups, then G has two canonical decompositions as a graph product of groups: a unique decomposition in which each vertex group is a directly indecomposable cyclic group, and a unique decomposition in which each vertex group is a finitely generated abelian group and the graph satisfies the T0 property. Our results build on results by Droms, Laurence and Radcliffe.

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 CONJUGATING AUTOMORPHISMS OF GRAPH PRODUCTS: KAZHDAN’S PROPERTY (T) AND SQ-UNIVERSALITY

2019 ◽  
Vol 101 (2) ◽  
pp. 272-282 ◽  
Author(s):  
ANTHONY GENEVOIS ◽  
OLGA VARGHESE
Keyword(s):  
Graph Products ◽  
Graph Product ◽  
Property T ◽  
Automorphism Of A Graph ◽  
Product Of Groups

An automorphism of a graph product of groups is conjugating if it sends each factor to a conjugate of a factor (possibly different). In this article, we determine precisely when the group of conjugating automorphisms of a graph product satisfies Kazhdan’s property (T) and when it satisfies some vastness properties including SQ-universality.

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.

Graph products of groups and group spaces

1995 ◽  
Vol 53 (1-2) ◽  
pp. 131-147 ◽  
Author(s):  
Jochen Pfalzgraf
Keyword(s):  
Graph Products ◽  
Products Of Groups
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