scholarly journals CSA-groups and separated free constructions

1995 ◽  
Vol 52 (1) ◽  
pp. 63-84 ◽  
Author(s):  
D. Gildenhuys ◽  
O. Kharlampovich ◽  
A. Myasnikov
Keyword(s):  
Abelian Subgroup ◽  
Dihedral Group ◽  
Hyperbolic Groups ◽  
Maximal Abelian Subgroup ◽  
Hnn Extensions ◽  
Abelian Subgroups ◽  
Amalgamated Products ◽  
Free Constructions ◽  
Torsion Free

A group G is called a CSA-group if all its maximal Abelian subgroups are malnormal; that is, Mx ∩ M = 1 for every maximal Abelian subgroup M and x ∈ G − M. The class of CSA-groups contains all torsion-free hyperbolic groups and all groups freely acting on λ-trees. We describe conditions under which HNN-extensions and amalgamated products of CSA-groups are again CSA. One-relator CSA-groups are characterised as follows: a torsion-free one-relator group is CSA if and only if it does not contain F2 × Z or one of the nonabelian metabelian Baumslag-Solitar groups B1, n = 〈x, y | yxy−1 = xn〉, n ∈ Z ∂ {0, 1}; a one-relator group with torsion is CSA if and only if it does not contain the infinite dihedral group.

scholarly journals Hyperbolic Groups are Hyperhopfian

2000 ◽  
Vol 68 (1) ◽  
pp. 126-130 ◽  
Author(s):  
Robert J. Daverman
Keyword(s):  
Abelian Subgroup ◽  
Hyperbolic Manifolds ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Residually Finite ◽  
Torsion Free

AbstractThe main result indicates that every finitely generated, residually finite, torsion-free, cohopfian group having on free Abelian subgroup of rank two is hyperhopfian. The argument relies on earlier work and ideas of Hirshon. As a corollary, fundamental groups of all closed hyperbolic manifolds are hyperhopfian.

Maximal Abelian Subgroups of the Symmetric Groups

1971 ◽  
Vol 23 (3) ◽  
pp. 426-438 ◽  
Author(s):  
John D. Dixon
Keyword(s):  
Symmetric Group ◽  
General Problem ◽  
Abelian Subgroup ◽  
Symmetric Groups ◽  
Specific Class ◽  
Maximal Abelian Subgroup ◽  
Number Of Generators ◽  
Abelian Subgroups ◽  
Almost All

Our aim is to present some global results about the set of maximal abelian subgroups of the symmetric group Sn. We shall show that certain properties are true for “almost all” subgroups of this set in the sense that the proportion of subgroups which have these properties tends to 1 as n → ∞. In this context we consider the order and the number of orbits of a maximal abelian subgroup and the number of generators which the group requires.Earlier results of this kind may be found in the papers [1; 2; 3; 4; 5]; the papers of Erdös and Turán deal with properties of the set of elements of Sn. The present work arose out of a conversation with Professor Turán who posed the general problem: given a specific class of subgroups (e.g., the abelian subgroups or the solvable subgroups) of Sn, what kind of properties hold for almost all subgroups of the class?

Amalgamated Products and the Howson Property

1997 ◽  
Vol 40 (3) ◽  
pp. 330-340 ◽  
Author(s):  
Ilya Kapovich
Keyword(s):  
Hyperbolic Group ◽  
Hyperbolic Groups ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Word Hyperbolic Group ◽  
Amalgamated Products ◽  
Word Hyperbolic Groups ◽  
Free Word ◽  
Torsion Free ◽  
Maximal Cyclic Subgroup

AbstractWe show that if A is a torsion-free word hyperbolic group which belongs to class (Q), that is all finitely generated subgroups of A are quasiconvex in A, then any maximal cyclic subgroup U of A is a Burns subgroup of A. This, in particular, implies that if B is a Howson group (that is the intersection of any two finitely generated subgroups is finitely generated) then A *UB, ⧼A, t | Ut = V⧽ are also Howson groups. Finitely generated free groups, fundamental groups of closed hyperbolic surfaces and some interesting 3-manifold groups are known to belong to class (Q) and our theorem applies to them. We also describe a large class of word hyperbolic groups which are not Howson.

scholarly journals CommutativeC⁎-Algebras of Toeplitz Operators via the Moment Map on the Polydisk

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Mauricio Hernández-Marroquin ◽  
Armando Sánchez-Nungaray ◽  
Luis Alfredo Dupont-García
Keyword(s):  
Abelian Subgroup ◽  
Toeplitz Operators ◽  
The Other ◽  
Moment Map ◽  
Maximal Abelian Subgroup ◽  
Other Hand ◽  
Algebras Of Toeplitz Operators ◽  
Abelian Subgroups ◽  
The Moment

We found that in the polydiskDnthere exist(n+1)(n+2)/2different classes of commutativeC⁎-algebras generated by Toeplitz operators whose symbols are invariant under the action of maximal Abelian subgroups of biholomorphisms. On the other hand, using the moment map associated with each (not necessary maximal) Abelian subgroup of biholomorphism we introduced a family of symbols given by the moment map such that theC⁎-algebra generated by Toeplitz operators with this kind of symbol is commutative. Thus we relate to each Abelian subgroup of biholomorphisms a commutativeC⁎-algebra of Toeplitz operators.

Maximal abelian subgroups of free profinite groups

1985 ◽  
Vol 97 (1) ◽  
pp. 51-55 ◽  
Author(s):  
Dan Haran ◽  
Alexander Lubotzky
Keyword(s):  
Group Theory ◽  
Abelian Subgroup ◽  
Profinite Group ◽  
Profinite Groups ◽  
Profinite Completion ◽  
Maximal Abelian Subgroup ◽  
Free Profinite Group ◽  
Abelian Subgroups

The aim of this note is to answer in the negative a question of W. -D. Geyer, asked at the 1983 Group Theory Meeting in Oberwolfach: Is a maximal abelian subgroup A of a free profinite group F necessarily isomorphic to , the profinite completion of

scholarly journals On infinite groups in which all abelian subgroups are locally cyclic

2021 ◽  
Author(s):  
Costantino Delizia ◽  
Chiara Nicotera
Keyword(s):  
Finite Groups ◽  
Abelian Subgroup ◽  
Periodic Case ◽  
Infinite Groups ◽  
Locally Finite ◽  
Locally Finite Groups ◽  
Periodic Groups ◽  
Abelian Subgroups

AbstractThe structure of locally soluble periodic groups in which every abelian subgroup is locally cyclic was described over 20 years ago. We complete the aforementioned characterization by dealing with the non-periodic case. We also describe the structure of locally finite groups in which all abelian subgroups are locally cyclic.

scholarly journals Quasi-isometric rigidity for graphs of virtually free groups with two-ended edge groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sam Shepherd ◽  
Daniel J. Woodhouse
Keyword(s):  
Free Group ◽  
Large Family ◽  
Finitely Generated ◽  
Free Vertex ◽  
Cyclic Subgroups ◽  
Graphs Of Groups ◽  
Jsj Decomposition ◽  
Finitely Generated Groups ◽  
Hnn Extensions ◽  
Abelian Subgroups

Abstract We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let G be a group that is one-ended, hyperbolic relative to virtually abelian subgroups, and has JSJ decomposition over two-ended subgroups containing only virtually free vertex groups that are not quadratically hanging. Our main result is that any group quasi-isometric to G is abstractly commensurable to G. In particular, our result applies to certain “generic” HNN extensions of a free group over cyclic subgroups.

Semistability of amalgamated products and HNN-extensions

1992 ◽  
Vol 98 (471) ◽  
pp. 0-0 ◽  
Author(s):  
Michael L. Mihalik ◽  
Steven T. Tschantz
Keyword(s):  
Hnn Extensions ◽  
Amalgamated Products

Abelian subgroup separability of certain HNN extensions

2018 ◽  
Vol 28 (03) ◽  
pp. 543-552
Author(s):  
Wei Zhou ◽  
Goansu Kim
Keyword(s):  
Abelian Subgroup ◽  
Nilpotent Groups ◽  
Finitely Generated ◽  
Hnn Extensions ◽  
Subgroup Separability

In this paper, we prove that certain HNN extensions of finitely generated abelian subgroup separable groups are finitely generated abelian subgroup separable. Using this, we show that certain HNN extensions of finitely generated nilpotent groups are finitely generated abelian subgroup separable.

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