The conjugacy problem for a free product with amalgamation

1971 ◽  
Vol 22 (1) ◽  
pp. 358-362 ◽  
Author(s):  
C. R. J. Clapham
Keyword(s):  
Free Product ◽  
Conjugacy Problem ◽  
Free Product With Amalgamation

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 A classification of the commutator subgroup of the group of a boundary link

1974 ◽  
Vol 18 (2) ◽  
pp. 216-221
Author(s):  
Bai Ching Chang
Keyword(s):  
Free Product ◽  
Free Group ◽  
Commutator Subgroup ◽  
Free Product With Amalgamation ◽  
Rank 2

In Neuwirth's book “Knot Groups” ([2]), the structure of the commutator subgroup of a knot is studied and characterized. Later Brown and Crowell refined Neuwith's result ([1], and we thus know that ifGis the groups of a knotK, then [G, G] is either free of rank 2g, wheregis the genus ofK, or a nontrivial free product with amalgamation on a free group of rank 2g, and may be written in the form, whereFis free of rank 2g, and the amalgamations are all proper and identical.

Lifting amalgamated sums and other colimits of groups and topological groups

1987 ◽  
Vol 102 (2) ◽  
pp. 273-280 ◽  
Author(s):  
Ronald Brown ◽  
Philip R. Heath
Keyword(s):  
Free Product ◽  
Topological Groups ◽  
Free Product With Amalgamation ◽  
Image Position

Suppose a group H is given as a free product with amalgamationdetermined by groups A0, A1, A2 and homomorphisms α1: A0 → A1, α2: A0 → A2. Thus H may be described as the quotient of the free product A * A2 by the relations i1 α1 (α0) = i2α2 (α0) for all α0 ∈ A0, where i1, i2 are the two injections of A1, A2 into A1 * A2. We do not assume that α1, α2 are injective, so the canonical homomorphisms α′i: Ai → H, i = 0,1,2, also need not be injective.

BICYCLIC SUBSEMIGROUPS IN AMALGAMS OF FINITE INVERSE SEMIGROUPS

2010 ◽  
Vol 20 (01) ◽  
pp. 89-113 ◽  
Author(s):  
EMANUELE RODARO
Keyword(s):  
Inverse Semigroup ◽  
Free Product ◽  
Sufficient Conditions ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Inverse Semigroups ◽  
Amalgamated Free Product ◽  
Free Product With Amalgamation ◽  
Necessary And Sufficient ◽  
Completely Semisimple

It is well known that an inverse semigroup is completely semisimple if and only if it does not contain a copy of the bicyclic semigroup. We characterize the amalgams [S1, S2; U] of two finite inverse semigroups S1, S2whose free product with amalgamation is completely semisimple and we show that checking whether the amalgamated free product of finite inverse semigroups contains a bicyclic subsemigroup is decidable by means of a polynomial time algorithm with respect to max {|S1|,|S2|}. Moreover we consider amalgams of finite inverse semigroups respecting the [Formula: see text]-order proving that the free product with amalgamation is completely semisimple and we also provide necessary and sufficient conditions for the [Formula: see text]-classes to be finite.

scholarly journals On the group ring of a free product with amalgamation

1980 ◽  
Vol 21 (1) ◽  
pp. 135-138
Author(s):  
Camilla R. Jordan
Keyword(s):  
Normal Subgroup ◽  
Free Product ◽  
Group Ring ◽  
Proper Subgroup ◽  
Finitely Generated ◽  
Free Product With Amalgamation

Let G = A*HB be the free product of the groups A and B amalgamating the proper subgroup H and let R be a ring with 1. If H is finite and G is not finitely generated we show that any non-zero ideal I of R(G) intersects non-trivially with the group ring R(M), where M = M(I) is a subgroup of G which is a free product amalgamating a finite normal subgroup. This result compares with A. I. Lichtman's results in [6] but is not a direct generalisation of these.

Free Products with Amalgamation and p-Adic Lie Groups

1998 ◽  
Vol 41 (4) ◽  
pp. 423-433
Author(s):  
D. D. Long ◽  
A. W. Reid
Keyword(s):  
Lie Groups ◽  
Free Product ◽  
Finitely Generated ◽  
Free Products ◽  
Finitely Generated Group ◽  
Free Product With Amalgamation ◽  
Free Products With Amalgamation

AbstractUsing the theory of p-adic Lie groups we give conditions for a finitely generated group to admit a splitting as a non-trivial free product with amalgamation. This can be viewed as an extension of a theorem of Bass.

scholarly journals Completely metrisable groups acting on trees

2011 ◽  
Vol 76 (3) ◽  
pp. 1005-1022
Author(s):  
Christian Rosendal
Keyword(s):  
Free Product ◽  
Finite Subgroup ◽  
Amplitude Function ◽  
Mild Conditions ◽  
Free Product With Amalgamation ◽  
Groups Acting On Trees

AbstractWe consider actions of completely metrisable groups on simplicial trees in the context of the Bass–Serre theory. Our main result characterises continuity of the amplitude function corresponding to a given action. Under fairly mild conditions on a completely metrisable groupG, namely, that the set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any decomposition as a free product with amalgamation,G=A*cB, the amalgamated groupsA,BandCare open inG.

scholarly journals Two-ended quasi-transitive graphs

2021 ◽  
Author(s):  
Babak Miraftab ◽  
Tim Rühmann
Keyword(s):  
Finite Group ◽  
Free Product ◽  
Hnn Extension ◽  
Free Product With Amalgamation ◽  
Finite Subgroups ◽  
A Finite Group ◽  
Transitive Graphs

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation [Formula: see text] or an HNN-extension [Formula: see text], where [Formula: see text] is a finite group and [Formula: see text] and [Formula: see text]. In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs.

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