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.

Right-Angled Artin Groups

2017 ◽  
Author(s):  
Robert W. Bell ◽  
Matt Clay
Keyword(s):  
Group Theory ◽  
Broad Spectrum ◽  
Geometric Group Theory ◽  
The Other ◽  
Artin Group ◽  
Artin Groups ◽  
Research Projects ◽  
Open Questions ◽  
Free Abelian Groups ◽  
Right Angled Artin Groups

This chapter deals with right-angled Artin groups, a broad spectrum of groups that includes free groups on one end, free abelian groups on the other end, and many other interesting groups in between. A right-angled Artin group is a group G(Γ‎) defined in terms of a graph Γ‎. Right-angled Artin groups have taken a central role in geometric group theory, mainly due to their involvement in the solution to one of the main open questions in the topology of 3-manifolds. The chapter first considers right-angled Artin groups as subgroups and how they relate to other classes of groups before exploring subgroups of right-angled Artin groups and the word problem for right-angled Artin groups. The discussion includes exercises and research projects.

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.

and Spaces

2017 ◽  
Author(s):  
Matt Clay ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Group Action ◽  
Metric Spaces ◽  
Geometric Group Theory ◽  
Geometric Object ◽  
Geometric Objects ◽  
Group Of Symmetries

This chapter discusses the notion of space, first by explaining what it means for a group to be a group of symmetries of a geometric object. This is the idea of group action, and some examples are given. The chapter proceeds by defining, for any group G, the Cayley graph of G and shows that the symmetric group of of this graph is precisely the group G. It then introduces metric spaces, which formalize the notion of a geometric object, and highlights numerous metric spaces that groups can act on. It also demonstrates that groups themselves are metric spaces; in other words, groups themselves can be thought of as geometric objects. The chapter concludes by using these ideas to frame the motivating questions of geometric group theory. Exercises relevant to each idea are included.

Quasi-isometries

2017 ◽  
Author(s):  
Dan Margalit ◽  
Anne Thomas
Keyword(s):  
Group Theory ◽  
Algebraic Structure ◽  
Cayley Graphs ◽  
Geometric Group Theory ◽  
Schwarz Lemma ◽  
Research Projects ◽  
Geometric Properties ◽  
Fundamental Lemma ◽  
Lipschitz Equivalence ◽  
Sample Result

This chapter considers the notion of quasi-isometry, also known as “coarse isometry.” A whole suite of important algebraic and geometric properties is preserved by quasi-isometries. Quasi-isometry can be applied to the algebraic structure of groups. A sample result, which shows that quasi-isometries can have powerful algebraic consequences, is a theorem of Gromov. Along the way to this theorem, the chapter proves the Milnor–Schwarz lemma, sometimes referred to as the fundamental lemma of geometric group theory. After describing Cayley graphs as well as path metrics and word metrics for integers, the chapter explores the bi-Lipschitz equivalence of word metrics, quasi-isometric equivalence of Cayley graphs, quasi-isometries between groups and spaces, and quasi-isometric rigidity. The discussion includes exercises and research projects.

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 Metric Thickenings and Group Actions

2020 ◽  
pp. 1-27
Author(s):  
Henry Adams ◽  
Mark Heim ◽  
Chris Peterson
Keyword(s):  
Group Theory ◽  
Homotopy Type ◽  
Metric Space ◽  
Metric Spaces ◽  
Scale Parameter ◽  
Orbit Space ◽  
Projective Spaces ◽  
Geometric Group Theory ◽  
Intermediate Scale ◽  
Scale Parameters

Let [Formula: see text] be a group acting properly and by isometries on a metric space [Formula: see text]; it follows that the quotient or orbit space [Formula: see text] is also a metric space. We study the Vietoris–Rips and Čech complexes of [Formula: see text]. Whereas (co)homology theories for metric spaces let the scale parameter of a Vietoris–Rips or Čech complex go to zero, and whereas geometric group theory requires the scale parameter to be sufficiently large, we instead consider intermediate scale parameters (neither tending to zero nor to infinity). As a particular case, we study the Vietoris–Rips and Čech thickenings of projective spaces at the first scale parameter where the homotopy type changes.

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.

Automorphisms of Free Groups

2017 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Symmetric Group ◽  
Cayley Graph ◽  
Free Group ◽  
Free Groups ◽  
Research Projects ◽  
Frobenius Theorem ◽  
Automorphisms Of Free Groups ◽  
Train Tracks

This chapter discusses the automorphisms of free groups. Every group is the collection of symmetries of some object, namely, its Cayley graph. A symmetry of a group is called an automorphism; it is merely an isomorphism of the group to itself. The collection of all of the automorphisms is also a group too, known as the automorphism group and denoted by Aut (G). The chapter considers basic examples of groups to illustrate what an automorphism is, with a focus on the automorphisms of the symmetric group on three elements and of the free abelian group. It also examines the dynamics of an automorphism of a free group and concludes with a description of train tracks, a topological model for the free group, and the Perron–Frobenius theorem. Exercises and research projects are included.

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