scholarly journals Congruences on the partial automorphism monoid of a free group action

2021 ◽  
pp. 1-30
Author(s):  
Matthew D. G. K. Brookes
Keyword(s):  
Free Group ◽  
Group Action ◽  
Finite Rank ◽  
Brandt Semigroup ◽  
Semidirect Products ◽  
Products Of Groups ◽  
Free Group Action

We study congruences on the partial automorphism monoid of a finite rank free group action. We determine a decomposition of a congruence on this monoid into a Rees congruence, a congruence on a Brandt semigroup and an idempotent separating congruence. The constituent parts are further described in terms of subgroups of direct and semidirect products of groups. We utilize this description to demonstrate how the number of congruences on the partial automorphism monoid depends on the group and the rank of the action.

Ends of Groups

2017 ◽  
Author(s):  
Nic Koban ◽  
John Meier
Keyword(s):  
Free Group ◽  
Group Action ◽  
Cantor Set ◽  
Braid Groups ◽  
Infinite Graphs ◽  
Research Projects ◽  
Semidirect Products ◽  
Number Of Ends ◽  
Ends Of Groups ◽  
Insight Into

This chapter focuses on the ends of a group. It first constructs a group action on the Cantor set and creates a free group from bijections of the Cantor set before showing how the idea of trying to understand what is happening at infinity for an infinite group is captured by the phrase “the ends of a group.” It then explores the notion of ends in the context of infinite graphs and presents examples that provide some insight into the number of ends of groups. It also looks at semidirect products and demonstrates how to calculate the number of ends of the braid groups before moving beyond the process of counting the ends of a group, taking into account the ends of the 4-valent tree. The discussion includes exercises and research projects.

scholarly journals On the generic type of the free group

2011 ◽  
Vol 76 (1) ◽  
pp. 227-234 ◽  
Author(s):  
Rizos Sklinos
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Free Groups ◽  
Primitive Elements ◽  
Generic Type

AbstractWe answer a question raised in [9], that is whether the infinite weight of the generic type of the free group is witnessed in Fω. We also prove that the set of primitive elements in finite rank free groups is not uniformly definable. As a corollary, we observe that the generic type over the empty set is not isolated. Finally, we show that uncountable free groups are not ℵ1-homogeneous.

scholarly journals A note on groups with separable finitely generated subgroups

1987 ◽  
Vol 36 (1) ◽  
pp. 153-160 ◽  
Author(s):  
R. G. Burns ◽  
A. Karrass ◽  
D. Solitar
Keyword(s):  
Free Group ◽  
Finite Rank ◽  
Cyclic Extension ◽  
Normal Subgroups ◽  
Finitely Generated ◽  
Positive Direction ◽  
One Relator Groups

An example is given of an infinite cyclic extension of a free group of finite rank in which not every finitely generated subgroup is finitely separable. This answers negatively the question of Peter Scott as to whether in all finitely generated 3-manifold groups the finitely generated subgroups are finitely separable. In the positive direction it is shown that in knot groups and one-relator groups with centre, the finitely generated normal subgroups are finitely separable.

scholarly journals ON THE COMPLEXITY OF THE WHITEHEAD MINIMIZATION PROBLEM

2007 ◽  
Vol 17 (08) ◽  
pp. 1611-1634 ◽  
Author(s):  
ABDÓ ROIG ◽  
ENRIC VENTURA ◽  
PASCAL WEIL
Keyword(s):  
Free Group ◽  
Polynomial Time ◽  
Minimization Problem ◽  
Finite Rank ◽  
Polynomial Algorithm ◽  
Minimum Size ◽  
Input Word ◽  
Finitely Generated ◽  
Factor Problem

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem — to decide whether a word is an element of some basis of the free group — and the free factor problem can also be solved in polynomial time.

Groups Acting on Trees

2017 ◽  
Author(s):  
Dan Margalit
Keyword(s):  
Free Group ◽  
Group Action ◽  
Research Projects ◽  
Nontrivial Element ◽  
Connected Graphs ◽  
Farey Tree ◽  
Groups Acting On Trees ◽  
Free Actions

This chapter considers groups acting on trees. It examines which groups act on which spaces and, if a group does act on a space, what it says about the group. These spaces are called trees—that is, connected graphs without cycles. A group action on a tree is free if no nontrivial element of the group preserves any vertex or any edge of the tree. The chapter first presents the theorem stating that: If a group G acts freely on a tree, then G is a free group. The condition that G is free is equivalent to the condition that G acts freely on a tree. The discussion then turns to the Farey tree and shows how to construct the Farey complex using the Farey graph. The chapter concludes by describing free and non-free actions on trees. Exercises and research projects are included.

Products of groups with finite rank

1987 ◽  
Vol 49 (5) ◽  
pp. 369-375 ◽  
Author(s):  
Bernhard Amberg
Keyword(s):  
Finite Rank ◽  
Products Of Groups

AUTOMORPHISMS OF RELATIVELY FREE GROUPS IN THE VARIETY N2A ∧ AN2 ∧ Nc

2001 ◽  
Vol 63 (3) ◽  
pp. 607-622 ◽  
Author(s):  
ATHANASSIOS I. PAPISTAS
Keyword(s):  
Automorphism Group ◽  
Free Group ◽  
Finite Index ◽  
Finite Rank ◽  
Free Groups ◽  
Finite Set ◽  
Tame Automorphism ◽  
Positive Integers ◽  
Relatively Free Group

For positive integers n and c, with n [ges ] 2, let Gn, c be a relatively free group of finite rank n in the variety N2A ∧ AN2 ∧ Nc. It is shown that the subgroup of the automorphism group Aut(Gn, c) of Gn, c generated by the tame automorphisms and an explicitly described finite set of IA-automorphisms of Gn, c has finite index in Aut(Gn, c). Furthermore, it is proved that there are no non-trivial elements of Gn, c fixed by every tame automorphism of Gn, c.

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