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.

scholarly journals Quantitative nonlinear embeddings into Lebesgue sequence spaces

2016 ◽  
Vol 08 (01) ◽  
pp. 117-150
Author(s):  
Florent P. Baudier
Keyword(s):  
Group Theory ◽  
Positive Solution ◽  
Metric Spaces ◽  
Exact Formula ◽  
Geometric Group Theory ◽  
Sequence Spaces ◽  
Text Compression ◽  
Property A ◽  
Locally Compact ◽  
Strong Embeddability

In this paper fundamental nonlinear geometries of Lebesgue sequence spaces are studied in their quantitative aspects. Applications of this work are a positive solution to the strong embeddability problem from [Formula: see text] into [Formula: see text] ([Formula: see text]) and new insights on the coarse embeddability problem from [Formula: see text] into [Formula: see text], [Formula: see text]. Relevant to geometric group theory purposes, the exact [Formula: see text]-compression of [Formula: see text] is computed. Finally coarse deformation of metric spaces with property A and locally compact amenable groups is investigated.

scholarly journals A ŠVARC–MILNOR LEMMA FOR MONOIDS ACTING BY ISOMETRIC EMBEDDINGS

2011 ◽  
Vol 21 (07) ◽  
pp. 1135-1147 ◽  
Author(s):  
ROBERT GRAY ◽  
MARK KAMBITES
Keyword(s):  
Group Theory ◽  
Cayley Graph ◽  
Semigroup Theory ◽  
Geometric Group Theory ◽  
Distance Functions ◽  
Cancellative Monoid ◽  
Isometric Embeddings ◽  
Key Techniques

We continue our program of extending key techniques from geometric group theory to semigroup theory, by studying monoids acting by isometric embeddings on spaces equipped with asymmetric, partially defined distance functions. The canonical example of such an action is a cancellative monoid acting by translation on its Cayley graph. Our main result is an extension of the Švarc–Milnor lemma to this setting.

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.

Groups

2017 ◽  
Author(s):  
Matt Clay ◽  
Dan Margalit
Keyword(s):  
Group Theory ◽  
Electric Field ◽  
Geometric Group Theory ◽  
Normal Subgroups ◽  
Infinite Groups ◽  
Higher Dimensional ◽  
Group Presentations ◽  
Planar Shape ◽  
Group Of Symmetries

This chapter considers the notion of a group in mathematics. It begins with a discussion of the problem of determining the symmetry of an object such as a planar shape, a higher-dimensional solid, a group, or an electric field. It then describes every group as a group of symmetries of some object and shows what it means for a group to be a group of symmetries of an object. These ideas are at the very heart of geometric group theory, the study of groups, spaces, and the interactions between them. The chapter also examines infinite groups, homomorphisms and normal subgroups, and group presentations. A number of exercises 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.

Geometric Group Theory 1991 Problem List

1993 ◽  
pp. 208-212
Author(s):  
Graham A. Niblo
Keyword(s):  
Group Theory ◽  
Geometric Group Theory ◽  
Problem List

scholarly journals ТЕОРИЈE РАЗВОЈA ГЕОМЕТРИЈСКОГ МИШЉЕЊА ПРЕМА ВАН ХИЛУ, ФИШБАЈНУ И УДЕМОН-КУЗНИАКУ

TEME ◽  
2017 ◽  
pp. 623 ◽  
Author(s):  
Оливера Ђокић ◽  
Маријана Зељић
Keyword(s):  
Theoretical Analysis ◽  
Mathematics Curriculum ◽  
Theoretical Framework ◽  
Geometric Object ◽  
Geometric Reasoning ◽  
Theoretical Frameworks ◽  
Teaching Geometry ◽  
Content Domain ◽  
Geometric Objects ◽  
Development Of Students

This research is a pedagogical study of theoretical frameworks of development of students’ geometrical thinking in various forms, particularly students’ geometric reasoning in teaching geometry: 1) model of van Hieles’ levels of understanding of geometry, 2) theory of figural concepts of Fischbein and 3) paradigms of Houdement-Kuzniak development of geometrical thinking. The aim of our research was to analyze the three theoretical framework and explain the reasons for their choice and expose them in terms of finding opportunities to permeate and connect them into one complete theory. The study used a descriptive-analytical and analytical-critical method of theoretical analysis. The results show that from each of the three theoretical frameworks we can clearly notice and distinguish geometric objects, as the students do not see them. They see them blended and structured in a series of procedures, and for that very reason we can say that they are poorly linked. We also opened questions for further research of geometric object as an important element for content domain geometry within mathematics curriculum.

Geometric Group Theory

2007 ◽  
Keyword(s):  
Group Theory ◽  
Geometric Group Theory
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