AUTOMATIC STRUCTURES FOR FUNDAMENTAL GROUPOIDS OF GRAPHS OF GROUPS

2005 ◽  
Vol 15 (01) ◽  
pp. 129-142
Author(s):  
PAUL LUNAU
Keyword(s):  
Free Product ◽  
Sufficient Conditions ◽  
Graph Of Groups ◽  
Automatic Structures ◽  
Graphs Of Groups ◽  
Free Product With Amalgamation ◽  
Fundamental Groupoid

We give sufficient conditions for the fundamental groupoid of a graph of groups to be automatic (resp. asynchronously automatic). These conditions are similar to those in [1] for a free product with amalgamation to be automatic (resp. asynchronously automatic).

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 AUTOMATIC STRUCTURES AND BOUNDARIES FOR GRAPHS OF GROUPS

1994 ◽  
Vol 04 (04) ◽  
pp. 591-616 ◽  
Author(s):  
WALTER D. NEUMANN ◽  
MICHAEL SHAPIRO
Keyword(s):  
Fundamental Group ◽  
Equivalence Relation ◽  
Infinite Product ◽  
Graph Product ◽  
Graph Of Groups ◽  
Automatic Structures ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Edge Group ◽  
Labelled Graphs

We study the synchronous and asynchronous automatic structures on the fundamental group of a graph of groups in which each edge group is finite. Up to a natural equivalence relation, the set of biautomatic structures on such a graph product bijects to the product of the sets of biautomatic structures on the vertex groups. The set of automatic structures is much richer. Indeed, it is dense in the infinite product of the sets of automatic structures of all conjugates of the vertex groups. We classify these structures by a class of labelled graphs which “mimic” the underlying graph of the graph of groups. Analogous statements hold for asynchronous automatic structures. We also discuss the boundaries of these structures.

scholarly journals Serre’s Property (FA) for automorphism groups of free products

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Naomi Andrew
Keyword(s):  
Automorphism Group ◽  
Free Product ◽  
Finite Groups ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Free Products ◽  
Cyclic Groups ◽  
Link Type ◽  
Open Case

AbstractWe provide some necessary and some sufficient conditions for the automorphism group of a free product of (freely indecomposable, not infinite cyclic) groups to have Property (FA). The additional sufficient conditions are all met by finite groups, and so this case is fully characterised. Therefore, this paper generalises the work of N. Leder [Serre’s Property FA for automorphism groups of free products, preprint (2018), https://arxiv.org/abs/1810.06287v1]. for finite cyclic groups, as well as resolving the open case of that paper.

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.

scholarly journals ON THE STRUCTURE OF SOME REDUCED AMALGAMATED FREE PRODUCT C*-ALGEBRAS

2011 ◽  
Vol 22 (02) ◽  
pp. 281-306 ◽  
Author(s):  
NIKOLAY A. IVANOV
Keyword(s):  
Free Product ◽  
Sufficient Conditions ◽  
Positive Cone ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Free Products ◽  
C Algebras ◽  
Amalgamated Free Product ◽  
Necessary And Sufficient

We study some reduced free products of C*-algebras with amalgamations. We give sufficient conditions for the positive cone of the K0 group to be the largest possible. We also give sufficient conditions for simplicity and uniqueness of trace. We use the latter result to give a necessary and sufficient condition for simplicity and uniqueness of trace of the reduced C*-algebras of the Baumslag–Solitar groups BS(m, n).

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.

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