At one point i jotted down notes and waffled about things i learnt about on a mathematical blog. The notes assume a basic knowledge of group theory, and metric and topological spaces. My intention in this text is to give a fairly gentle introduction to several of the main methods and theorems in the subject. On the one hand, geometric group theory is an interesting theory combining aspects of di erent elds of mathematics in a cunning way. Milne top these are full notes for all the advanced graduatelevel courses i have taught since 1986. These connections lead to a fruitful interplay between geometry, topology and geometric group theory. This threeday workshop will feature talks by six prominent female mathematicians on a wide range of topics in geometric group theory. Some of walter neumanns preprints columbia university. Roller these two volumes contain survey papers given at the 1991 international symposium on geometric group theory, and they represent some of the latest thinking in this area. A comprehensive treatment to hyperbolic groups among.
Download dvi download pdf a course on geometric group theory brian h. Geometric group theory lives between algebra and topology group theory is the study of groups, which weve seen a few times before, and geometric means that well be looking at shapes. The geometric group theory page provides information and resources about geometric group theory and lowdimensional topology, although the links sometimes stray into neighboring fields. This book is an introduction into geometric group theory. Any vector space is a group with respect to the operation of vector addition. There are three particular reasons that make me feel this way. It is my intention one day to expand the notes to take account of this, and to produce a volume that, while still modest in size c200 pages, will provide a more comprehensive introduction to group theory for beginning graduate students in mathematics, physics, and related fields. Part iv topics in geometric group theory based on lectures by h. In a masters course, it is not possible to assume that all the students have reached any given level of pro. Geometry and group theory abstract inthiscourse, wedevelopthebasicnotionsofmanifoldsandgeometry, withapplications in physics, and also we develop the basic notions of. The selection of topics is biased by my own preferences, but i hope that this book will prepare and encourage the reader to discover more of geometric group theory. Geometric group theory attacks these problems from a geometric viewpoint, either by viewing groups as geometric objects, or by finding suitable geometric objects a group acts on. Introduction to geometric group theory andrzej nagorko.
Section 1 this is background and should be familiar from the previous group theory course. Lecture notes on geometric group theory shengkui ye shengkui. It strikes me now that while i love books i really do, i often find that i learn best from sets of lecture notes and short articles. Remark epstein, sela there is an algorithmic procedure to determine the number of ends i. When i later learned that jim shares my manx roots, charm was added to beauty. Geometric group theory is really a collection of things put together by a state of mind. As next step, we will introduce a metric structure on groups via word metrics on cayley graphs, and we will study the large scale geometry. Geometric group theory ggt for short uses geometric topological methods and ideas to come to conclusions about groups associated with shapes. Groups developed in parallel as part of number theoretic and geometric investigations due to euler, lagrange, cauchy and abel. The goal of this book is to present several central topics in geometric group theory, primarily related to the large scale geometry of infinite groups and spaces on which such groups act, and to illustrate them with fundamental theorems such as gromovs theorem on groups of polynomial growth. They generally take time to bring to the readers attention.
These lecture notes are the manuscript of a one semester course on geomet ric group theory, taught in 20102011. Names in parentheses re ect the person i heard the question. We present a brief overview of methods and results in geometric group theory, with the goal of introducing the reader to both topological and metric perspectives. Hull 1 introduction one of the main themes of geometric group theory is to study a nitely generated group gin terms of the geometric properties of the cayley graph of g. Notes for revision 1 whats examinable this is a rough guide to the most important parts of the module to concentrate on for revision. These lecture notes are the manuscript of a onesemester course on geomet ric group theory, taught in 20102011. Acknowledgements i thank the following for providing corrections and comments for earlier versions of these notes. These notes are an edited version of notes and transparencies i prepared. Geometric group theory is a vast, rapidly growing area of mathematics. Our goal this semester is to look as some speci c quasi. The term free abelian should be thought of as one word. This b o ok is based upon a set lecture notes for a course that i w as. Geometric group theory studies a group using topology and geometry.
Introduction to geometric group theory notes from course given by dick canary, fall 2014. First, one nds algorithmically an explicit of hyperbolicity. Lengths 3 1 word lengths and cayley graphs 3 2 the fundamental theorem of geometric group theory 4 3 classications of groups up to quasiisometries 6 ii lecture 2. Its origins lie in geometry where groups describe in a very detailed way the symmetries of geometric objects and in the theory. Carters group explorerherefor exploring the structure of groups of small order. Alessandro sisto, lecture notes on geometric group theory.
Geometry and group theory abstract in this course, we develop the basic notions of manifoldsandgeometry, with applications. From math 697, taught by richard canary at michigan in fall 2015. For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library. These can arise in all dimensions, but since we are constrained to working with 2dimensional paper, blackboards and computer screens, i will stick to 2dimensional examples. Note that science direct is a subscription service, and you must be connecting from a url in the tamu. Gromovs theorem and its corollary will be proven in chapter 14. Each speaker will give two lectures, separated by a breakout session during which participants will meet in small groups to discuss ideas presented in the first lecture. August 28, 2018 metric invariants and quantitative topology, infinite groups, locally symmetric spaces and beyond. On the other hand, geometric group theory has numerous applications to problems in classical elds such as group theory and riemannian geometry. Geometric group theory, such as coarse topology, ultralimits and quasiconformal.
The way to use geometry to study groups is considering their isometric actions on metric spaces. Free groups theory books download ebooks online textbooks. These \ geometric properties come in the form of quasiisometry invariants. Essays on geometric group theory international press. Each section is followed by a series of problems, partly to check understanding marked with the letter \r. There is no attempt to be encyclopedic, and proofs are usually only sketched, or only carried out for an illustrative special case. Representation theory university of california, berkeley. Cambridge notes below are the notes i took during lectures in cambridge, as well as the example sheets. This page is meant to help students, scholars, and interested laypersons orient themselves to this large and everexpanding body of work.
In both case we have transformations that help us to capture the type of symmetry we are interested in. The mathematical sciences research institute msri, founded in 1982, is an independent nonprofit mathematical research institution whose funding sources include the national science foundation, foundations, corporations, and more than 90 universities and institutions. Recommended problem, partly to present further examples or to extend theory. Within geometric group theory, the close relationships between mapping. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to. Wilton notes taken by dexter chua michaelmas 2017 these notes are not endorsed by the lecturers, and i. Topics in geometric group theory sameer kailasa abstract.
Geometric group theory is the art of studying groups without using algebra. To illustrate this we will look at two very di erent kinds of symmetries. Geometric group theory preliminary version under revision. For instance, a topological model of might be a space xwhose fundamental group. The aim of this course is to introduce the fundamental methods and problems of geometric group theory and discuss their relationship to topology and geometry. Let denote an equilateral triangle in the plane with origin as the centroid. Some of the notes give complete proofs group theory, fields and galois theory, algebraic number theory, class field theory, algebraic geometry, while others are more in the nature of introductory overviews to a topic. Note that the dot product u v is not a binary operation, since it. Preface the goal of this book is to present several central topics in geometric group theory,primarilyrelatedtothelargescalegeometryofin. Notes from a msri summer graduate school, summer 2015. We describe group presentations from first principles. Elements of group theory 1 lecture 0 wednesday, thursday, and friday, august 2628 1 a. Shape theory is an area of topologydeveloped for studying compact spaces with bad local properties, so it may seem odd that shapes is one of three topics mentioned. Representation theory reverses the question to given a group g, what objects x does it act on.
It is certainly not an encyclopedic treatment of geometric group theory, but hopefully it. Geometric group theory ggt for short uses geometrictopological methods and ideas to come to conclusions about groups associated with shapes. It more or less describes the minimum you should know for the exam. Georgi, lie algebras and particle physics, perseus books group. Minimal singularities are lipschitz normally embedded. The geometric concepts underlying this dictionary include the behaviour of elliptic. The mapping class group is connected to many areas of mathematics, including complex analysis, dynamics, algebraic geometry, algebraic topology, geometric topology particularly in the study of 3 and 4dimensional spaces, and group theory. It has arisen out of notes for courses given at the secondyear graduate level at the university of minnesota. Suppose that gis a group quasiisometric to a nilpotent group. Important examples of groups arise from the symmetries of geometric objects. The topology, geometry, and dynamics of free groups.
Pdf a course on geometric group theory semantic scholar. Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act that is, when the groups in question are realized as geometric symmetries or continuous transformations of. Galois himself used the word group as we understand it now. Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces. The origin of abstract group theory goes however further back to galois 18111832 and the problem of solving polynomial. Essays on geometric group theory proceedings of the instructional workshop and international conference on group theory, indian institute of technology, guwahati, december 2002. Modern study of groups has evolved into the eld of geometric group theory which thrives on a dialogue between algebra and geometry. Geometric group theory lecture notes university of florida. Geometric group theory cornelia drutu and michael kapovich with an appendix by bogdan nica. As next step, we will introduce a metric structure on groups via word metrics on cayley graphs, and we will study the large scale geometry of groups with respect to this metric structure. Then one nds an automatic structure, from which it can be immediately read if the group is nite or 2ended. A geometric model might be obtained by choosing some geometric structure on x, and then lifting that structure to the universal cover xeto obtain a deck action y xe.
The standard resources for geometric group theory are. Introduction to geometric group theory january 10, 2016. Geometric group theory lecture notes semantic scholar. These are lecture notes for the 2014 course on geometric group theory at eth zurich. They generally take time to bring to the readers attention some interesting side fact that would normally be left out of a standard textbook lest it be too big. Shapiro notes for the anu workshop januaryfebruary 1996.
Draft lecture notes for course 311 abstract algebra, taught at trinity college, dublin, in the academic year 200708, are available here. We describe some of the fundamental notions of geometric group. One studies the structure of groups, in particular the subgroup structure. This book is written for students who are studying nite group representation theory beyond the level of a rst course in abstract algebra. The first idea is made precise by means of the cayley graph, whose vertices correspond to group elements and edges correspond to right multiplication in the group. The articles in these two volumes arose from papers given at the 1991 international symposium on geometric group theory, and they represent some of the latest thinking in this area. The first part of the course begins with an introduction to presentations and the list of problems of m. Lectures on geometric group theory uc davis mathematics. Group theory can be viewed as the mathematical theory that deals with symmetry, where symmetry has a very general meaning.
Mark jankins and walter d neumann brandeis lecture notes 2, march 1983. Note that milnor stated the problem in the form of a question, not a conjec. Its most widely recognized use, however, is to describe the sub. Notes on group theory 5 here is an example of geometric nature. Major revision august 22, 2000 updated july 2004 notes. Lectures on groups and their connections to geometry. The institute is located at 17 gauss way, on the university of california, berkeley campus, close to grizzly peak, on the. A geometric introduction to ktheory daniel dugger contents preface 3 introduction 4 1. Warsaw university june 24th, 2007 an introductory talk about basic notions of geometric group theory, with emphasis on asymptotic invariants of groups. From this point of view, geometry asks, given a geometric object x, what is its group of symmetries. Note that science direct is a subscription service, and you must be connecting from. Course 311 abstract algebra lecture notes for the academic year 200708. An introduction to geometric group theory pristina matthieu dussaule mars 2017 this is a eight hours course that i gave at the university of pristina. We describe some of the fundamental notions of geometric group theory, such as quasiisometries, and aim for a basic overview of hyperbolic groups.
Wilton notes taken by dexter chua michaelmas 2017 these notes are not endorsed by the lecturers, and i have modi ed them often. Lecture notes in group theory gunnar traustason autumn 2016 0. This subject goes back to the early 20th century work of max dehn on surface groups, and in turn has its roots in late 19th century work on. Lectures on groups and their connections to geometry anatole katok vaughn climenhaga department of mathematics, pennsylvania state university. This is quite a useful introduction to some of the basics of lie. A course on geometric group theory uc davis mathematics.
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