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.

The Ping-Pong Lemma

2017 ◽  
Author(s):  
Johanna Mangahas
Keyword(s):  
Group Theory ◽  
Free Group ◽  
Group Action ◽  
Metric Space ◽  
Hyperbolic Geometry ◽  
Free Groups ◽  
Central Place ◽  
Geometric Group Theory ◽  
Research Projects ◽  
Ping Pong

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, you can tell a group by how it acts. That is, a good group action (for example, action by isometries on a metric space) can reveal a lot about the group itself. This theme occupies a central place in geometric group theory. The ping-pong lemma, also dubbed Schottky lemma or Klein's criterion, gives a set of circumstances for identifying whether a group is a free group. The chapter first presents the statement, proof, and first examples using ping-pong before discussing ping-pong with Möbius transformations and hyperbolic geometry. Exercises and research projects are included.

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.

Uniqueness of Free Actions on S3 Respecting a Knot

1987 ◽  
Vol 39 (4) ◽  
pp. 969-982 ◽  
Author(s):  
Michel Boileau ◽  
Erica Flapan
Keyword(s):  
Fixed Point ◽  
Cyclic Group ◽  
Group Action ◽  
Cyclic Groups ◽  
Finite Cyclic Group ◽  
Cyclic Group Action ◽  
Periodic Diffeomorphisms ◽  
Free Actions

In this paper we consider free actions of finite cyclic groups on the pair (S3, K), where K is a knot in S3. That is, we look at periodic diffeo-morphisms f of (S3, K) such that fn is fixed point free, for all n less than the order of f. Note that such actions are always orientation preserving. We will show that if K is a non-trivial prime knot then, up to conjugacy, (S3, K) has at most one free finite cyclic group action of a given order. In addition, if all of the companions of K are prime, then all of the free periodic diffeo-morphisms of (S3, K) are conjugate to elements of one cyclic group which acts freely on (S3, K). More specifically, we prove the following two theorems.THEOREM 1. Let K be a non-trivial prime knot. If f and g are free periodic diffeomorphisms of (S3, K) of the same order, then f is conjugate to a power of g.

scholarly journals Cohomology groups invariant under continuous orbit equivalence

2020 ◽  
pp. 1-35
Author(s):  
Yongle Jiang
Keyword(s):  
Group Action ◽  
Continuous Group ◽  
Orbit Equivalence ◽  
Cohomology Groups ◽  
Bounded Cohomology ◽  
Homology Groups ◽  
Free Actions ◽  
Uniformly Finite Homology

By the work of Brodzki–Niblo–Nowak–Wright and Monod, topological amenability of a continuous group action can be characterized using uniformly finite homology groups or bounded cohomology groups associated to this action. We show that (certain variations of) these groups are invariants for topologically free actions under continuous orbit equivalence.

Thompson’s Group

2017 ◽  
Author(s):  
Sean Cleary
Keyword(s):  
Group Action ◽  
Word Length ◽  
Research Projects ◽  
Direct Sum ◽  
Group Presentations ◽  
Thompson’S Group ◽  
Rank 2 ◽  
Definition Of ◽  
Finitely Presented ◽  
Length Distortion

This chapter considers the Thompson's group F. Thompson's group F exhibits several behaviors that appear paradoxical. For example: F is finitely presented and contains a copy of F x F, indicating that F contains the direct sum of infinitely many copies of F. In addition, F has exponential growth but contains no free groups of rank 2. After providing an overview of the analytic definition and basic properties of the Thompson's group, the chapter introduces a combinatorial definition of F and two group presentations for F, an infinite one and a finite one. It also explores the subgroups, quotients, endomorphisms, and group action of F before concluding with an analysis of several geometric properties of F such as word length, distortion, dead ends, and growth. The discussion includes exercises and research projects.

scholarly journals Incomparable actions of free groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2084-2098
Author(s):  
CLINTON T. CONLEY ◽  
BENJAMIN D. MILLER
Keyword(s):  
Probability Measure ◽  
Free Group ◽  
Equivalence Relation ◽  
Polish Space ◽  
Borel Probability Measure ◽  
Free Groups ◽  
Borel Equivalence Relation ◽  
Separability Condition ◽  
Borel Probability ◽  
Free Actions

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

Free Groups and Folding

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Normal Subgroup ◽  
Free Group ◽  
Finite Index ◽  
Free Groups ◽  
Membership Problem ◽  
Research Projects ◽  
Residually Finite ◽  
Simple Operation ◽  
Nontrivial Normal Subgroup ◽  
Subgroup Theorem

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show the rank of the free group H and whether every finitely generated nontrivial normal subgroup of a free group has finite index. The edge paths and the fundamental group of a graph are discussed, along with subgroups via graphs. The chapter also considers five applications of folding: the Nielsen–Schreier Subgroup theorem, the membership problem, index, normality, and residual finiteness. A group G is residually finite if for every nontrivial element g of G there is a normal subgroup N of finite index in G so that g is not in N. Exercises and research projects are included.

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.

scholarly journals On centralizers of elements of groups acting on trees with inversions

2003 ◽  
Vol 2003 (20) ◽  
pp. 1241-1249
Author(s):  
R. M. S. Mahmood
Keyword(s):  
Nontrivial Element ◽  
Groups Acting On Trees

A subgroupHof a groupGis called malnormal inGif it satisfies the condition that ifg∈Gandh∈H,h≠1such thatghg−1∈H, theng∈H. In this paper, we show that ifGis a group acting on a treeXwith inversions such that each edge stabilizer is malnormal inG, then the centralizerC(g)of each nontrivial elementgofGis in a vertex stabilizer ifgis in that vertex stabilizer. Ifgis not in any vertex stabilizer, thenC(g)is an infinite cyclic ifgdoes not transfer an edge ofXto its inverse. Otherwise,C(g)is a finite cyclic of order 2.

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