scholarly journals Finitely generated subgroups of amalgamated free products and HNN groups

1976 ◽  
Vol 22 (3) ◽  
pp. 274-281 ◽  
Author(s):  
Daniel E. Cohen
Keyword(s):  
Free Product ◽  
Finite Index ◽  
Subnormal Subgroup ◽  
Finitely Generated ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Hnn Extension ◽  
Groups Acting On Trees

AbstractThe theory of groups acting on trees due to Bass and Serre (1969) is applied to simplify some results of Burns (1972, 1973) giving conditions under which an amalgamated free product or HNN extension has the properties that any finitely generated subgroup containing an infinite subnormal subgroup must have finite index and that the intersection of two finitely generated subgroups is finitely generated.

Groups in which every Finitely Generated Subgroup is almost a Free Factor

1979 ◽  
Vol 31 (6) ◽  
pp. 1329-1338 ◽  
Author(s):  
A. M. Brunner ◽  
R. G. Burns
Keyword(s):  
Free Product ◽  
Free Group ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Products

In [5] M. Hall Jr. proved, without stating it explicitly, that every finitely generated subgroup of a free group is a free factor of a subgroup of finite index. This result was made explicit, and used to give simpler proofs of known results, in [1] and [7]. The standard generalization to free products was given in [2]: If, following [13], we call a group in which every finitely generated subgroup is a free factor of a subgroup of finite index an M. Hall group, then a free product of M. Hall groups is again an M. Hall group. The recent appearance of [13], in which this result is reproved, and the rather restrictive nature of the property of being an M. Hall group, led us to attempt to determine the structure of such groups. In this paper we go a considerable way towards achieving this for those M. Hall groups which are both finitely generated and accessible.

Second Order Dehn Functions for Amalgamated Free Products of Groups

2009 ◽  
Vol 16 (04) ◽  
pp. 699-708
Author(s):  
Xiaofeng Wang ◽  
Xiaomin Bao
Keyword(s):  
Free Product ◽  
Upper Bound ◽  
Second Order ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Products Of Groups ◽  
Amalgamated Free Product ◽  
Finite Set

A finite set of generators for a free product of two groups of type F3with a subgroup amalgamated, and an estimation for the upper bound of the second order Dehn functions of the amalgamated free product are carried out.

scholarly journals Radial multipliers on amalgamated free products of II1-factors

2014 ◽  
Vol 25 (03) ◽  
pp. 1450026
Author(s):  
Sören Möller
Keyword(s):  
Free Product ◽  
Von Neumann Algebras ◽  
Hankel Matrix ◽  
Trace Class ◽  
Free Products ◽  
Von Neumann ◽  
Amalgamated Free Products ◽  
Amalgamated Free Product ◽  
Radial Multipliers ◽  
Completely Bounded Map

Let ℳi be a family of II1-factors, containing a common II1-subfactor 𝒩, such that [ℳi : 𝒩] ∈ ℕ0 for all i. Furthermore, let ϕ: ℕ0 → ℂ. We show that if a Hankel matrix related to ϕ is trace-class, then there exists a unique completely bounded map Mϕ on the amalgamated free product of the ℳi with amalgamation over 𝒩, which acts as a radial multiplier. Hereby, we extend a result of Haagerup and the author for radial multipliers on reduced free products of unital C*- and von Neumann algebras.

scholarly journals Finitely generated subgroups and the centre of some factor groups of free products

1999 ◽  
Vol 60 (2) ◽  
pp. 239-244
Author(s):  
C.K. Gupta ◽  
N. Romanovski
Keyword(s):  
Free Product ◽  
Subnormal Subgroup ◽  
Finitely Generated ◽  
Free Products

For groups of the type F/[R, R], where F is a free product, we prove a generalisation of a theorem of Karrass and Solitar on a finitely generated subgroup of a free product containing a nontrivial subnormal subgroup. We also describe the centre of the group F[/R, R].

scholarly journals ON THE GROWTH OF GROUPS AND AUTOMORPHISMS

2005 ◽  
Vol 15 (05n06) ◽  
pp. 869-874 ◽  
Author(s):  
MARTIN R. BRIDSON
Keyword(s):  
Finitely Generated ◽  
Free Products ◽  
Dehn Functions ◽  
Amalgamated Free Products ◽  
Growth Functions ◽  
Growth Of Groups

We consider the growth functions βΓ(n) of amalgamated free products Γ = A *C B, where A ≅ B are finitely generated, C is free abelian and |A/C| = |A/B| = 2. For every d ∈ ℕ there exist examples with βΓ(n) ≃ nd+1βA(n). There also exist examples with βΓ(n) ≃ en. Similar behavior is exhibited among Dehn functions.

On Certain Finitely Generated Subgroups of Groups Which Split

2003 ◽  
Vol 46 (1) ◽  
pp. 122-129 ◽  
Author(s):  
Myoungho Moon
Keyword(s):  
Positive Integer ◽  
Free Product ◽  
Finite Index ◽  
Finitely Generated ◽  
Free Product With Amalgamation

AbstractDefine a group G to be in the class 𝒮 if for any finitely generated subgroup K of G having the property that there is a positive integer n such that gn ∈ K for all g ∈ G, K has finite index in G. We show that a free product with amalgamation A *CB and an HNN group A *C belong to 𝒮, if C is in 𝒮 and every subgroup of C is finitely generated.

scholarly journals EQUIVARIANT COMPRESSION OF CERTAIN DIRECT LIMIT GROUPS AND AMALGAMATED FREE PRODUCTS

2016 ◽  
Vol 58 (3) ◽  
pp. 739-752
Author(s):  
CHRIS CAVE ◽  
DENNIS DREESEN
Keyword(s):  
Finite Index ◽  
Direct Limit ◽  
Free Products ◽  
Limit Groups ◽  
Amalgamated Free Products

AbstractWe give a means of estimating the equivariant compression of a group G in terms of properties of open subgroups Gi ⊂ G whose direct limit is G. Quantifying a result by Gal, we also study the behaviour of the equivariant compression under amalgamated free products G1∗HG2 where H is of finite index in both G1 and G2.

scholarly journals Free products of topological groups with a closed subgroup amalgamated

1986 ◽  
Vol 40 (3) ◽  
pp. 414-420 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Free Product ◽  
Closed Subgroup ◽  
Countable Family ◽  
Topological Groups ◽  
Free Products ◽  
Dense Image ◽  
Amalgamated Free Product

AbstractIt is shown that if {Gn: n = 1, 2,…} is a countable family of Hausdorff kω-topological groups with a common closed subgroup A, then the topological amalgamated free product *AGn exists and is a Hausdorff kω-topological group with each Gn as a closed subgroup. A consequence is the theorem of La Martin that epimorphisms in the category of kω-topological groups have dense image.

scholarly journals C*-SIMPLE GROUPS: AMALGAMATED FREE PRODUCTS, HNN EXTENSIONS, AND FUNDAMENTAL GROUPS OF 3-MANIFOLDS

2011 ◽  
Vol 03 (04) ◽  
pp. 451-489 ◽  
Author(s):  
PIERRE DE LA HARPE ◽  
JEAN-PHILIPPE PRÉAUX
Keyword(s):  
Sufficient Conditions ◽  
Simple Groups ◽  
Fundamental Groups ◽  
Normal Subgroups ◽  
Free Products ◽  
Amalgamated Free Products ◽  
Hnn Extensions ◽  
Groups Acting On Trees ◽  
Jsj Decompositions ◽  
Fixed Point Sets

We establish sufficient conditions for the C*-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their nontrivial subnormal subgroups; for example normal subgroups of Baumslag–Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser–Milnor and JSJ decompositions. Much of our analysis deals with conditions on an action of a group Γ on a tree T which imply the following three properties: abundance of hyperbolic elements, better called strong hyperbolicity, minimality, both on the tree T and on its boundary ∂T, and faithfulness in a strong sense. An important step in this analysis is to identify automorphisms of T which are slender, namely such that their fixed-point sets in ∂T are nowhere dense for the shadow topology.

scholarly journals Markov semigroups, monoids and groups

2014 ◽  
Vol 24 (05) ◽  
pp. 609-653 ◽  
Author(s):  
Alan J. Cain ◽  
Victor Maltcev
Keyword(s):  
Finite Index ◽  
Regular Language ◽  
Minimal Length ◽  
Direct Products ◽  
Markov Semigroups ◽  
Finitely Generated ◽  
Free Products ◽  
Commutative Semigroups ◽  
Generating Set ◽  
Polycyclic Groups

A group is Markov if it admits a prefix-closed regular language of unique representatives with respect to some generating set, and strongly Markov if it admits such a language of unique minimal-length representatives over every generating set. This paper considers the natural generalizations of these concepts to semigroups and monoids. Two distinct potential generalizations to monoids are shown to be equivalent. Various interesting examples are presented, including an example of a non-Markov monoid that nevertheless admits a regular language of unique representatives over any generating set. It is shown that all finitely generated commutative semigroups are strongly Markov, but that finitely generated subsemigroups of virtually abelian or polycyclic groups need not be. Potential connections with word-hyperbolic semigroups are investigated. A study is made of the interaction of the classes of Markov and strongly Markov semigroups with direct products, free products, and finite-index subsemigroups and extensions. Several questions are posed.

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