Topics in Geometric Group Theory

Author: Pierre de la Harpe
Publisher: University of Chicago Press
ISBN: 9780226317199
Category: Mathematics
Page: 310
View: 8276

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In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe adopts a hands-on approach, illustrating key concepts with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing way, with an emphasis on finitely generated versus finitely presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group." Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields.

Geometrische Gruppentheorie

Ein Einstieg mit dem Computer Basiswissen für Studium und Mathematikunterricht
Author: Stephan Rosebrock
Publisher: Springer-Verlag
ISBN: 9783834810380
Category: Mathematics
Page: 211
View: 2099

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In diesem Buch geht es um Gruppentheorie. Man kann Gruppen als algebraische Objekte auffassen, die die Symmetrie von geometrischen Objekten beschreiben. Dieser Blickwinkel steht bei dem Buch im Vordergrund und somit geht es in dem Buch auch um Geometrie. Gruppen drücken Symmetriephänomene algebraisch aus, man rechnet mit Spiegelungen, Drehungen usw., allgemein mit Abbildungen von Räumen auf sich. Das Buch kann vorlesungsbegleitend bei Algebra- und Gruppentheorie-Vorlesungen eingesetzt werden. Es eignet sich auch besonders gut für Lehramtsstudierende, da es den Stoff computerorientiert (unter Benutzung des frei erhältlichen Gruppentheorie-Programms GAP) mit vielen anschaulichen Beispielen präsentiert. Für die 2. Auflage wurden einige Teile des Buches ausführlicher dargestellt. Einige Inhalte (zum Beispiel die Klassifikation der endlichen Gruppen bis zur Ordnung 11) wurden hinzugefügt und Fehler korrigiert. Einführung in die euklidische Geometrie – Einführung in Gruppen – Untergruppen und Homomorphismen – Gruppenoperationen – Gruppenpräsentationen – Produkte von Gruppen – Endliche Gruppen – Die hyperbolische Ebene – Hyperbolische Gruppen - Studierende der Mathematik und Naturwissenschaften ab dem 3. Semester - Studierende der Lehrämter - Mathematiklehrerinnen und -lehrer - Dozenten der Mathematik und ihrer Didaktik Akad. Oberrat Dr. Stephan Rosebrock ist Dozent für Mathematik an der PH Karlsruhe, Institut für Mathematik und Informatik.

Computational and Statistical Group Theory

AMS Special Session Geometric Group Theory, April 21-22, 2001, Las Vegas, Nevada, AMS Special Session Computational Group Theory, April 28-29, 2001, Hoboken, New Jersey
Author: Esther G Forbes,Robert H. Gilman,Alexei G. Myasnikov,Vladimir Shpilrain,Nev.) AMS Special Session Geometric Group Theory (2001 : Las Vegas
Publisher: American Mathematical Soc.
ISBN: 0821831585
Category: Mathematics
Page: 124
View: 1521

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This book gives a nice overview of the diversity of current trends in computational and statistical group theory. It presents the latest research and a number of specific topics, such as growth, black box groups, measures on groups, product replacement algorithms, quantum automata, and more. It includes contributions by speakers at AMS Special Sessions at The University of Nevada (Las Vegas) and the Stevens Institute of Technology (Hoboken, NJ). It is suitable for graduate students and research mathematicians interested in group theory.

Geometric Group Theory

Author: Cornelia Druţu,Michael Kapovich
Publisher: American Mathematical Soc.
ISBN: 1470411040
Category: Geometric group theory
Page: 819
View: 6208

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The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamental groups of manifolds, groups of matrices with integer coefficients, etc. The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls. The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the first book in which geometric group theory is presented in a form accessible to advanced graduate students and young research mathematicians. It fills a big gap in the literature and will be used by researchers in geometric group theory and its applications.

Office Hours with a Geometric Group Theorist

Author: Matt Clay,Dan Margalit
Publisher: Princeton University Press
ISBN: 1400885396
Category: Mathematics
Page: 456
View: 905

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Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Groups, Geometry and Dynamics

Author: N.A
Publisher: N.A
Category: Group theory
Page: N.A
View: 4569

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Publishes research articles that focus on groups or group actions as well as articles in other areas of mathematics in which groups or group actions are used as a main tool. Covers all topics of modern group theory with preference given to geometric, asymptotic and combinatorial group theory, dynamics of group actions, probabilistic and analytical methods, interaction with ergodic theory and operator algebras, and other related fields.

Lectures on Exceptional Lie Groups

Author: J. F. Adams
Publisher: University of Chicago Press
ISBN: 9780226005270
Category: Mathematics
Page: 122
View: 2703

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J. Frank Adams was internationally known and respected as one of the great algebraic topologists. Adams had long been fascinated with exceptional Lie groups, about which he published several papers, and he gave a series of lectures on the topic. The author's detailed lecture notes have enabled volume editors Zafer Mahmud and Mamoru Mimura to preserve the substance and character of Adams's work. Because Lie groups form a staple of most mathematics graduate students' diets, this work on exceptional Lie groups should appeal to many of them, as well as to researchers of algebraic geometry and topology. J. Frank Adams was Lowndean professor of astronomy and geometry at the University of Cambridge. The University of Chicago Press published his Lectures on Lie Groups and has reprinted his Stable Homotopy and Generalized Homology. Chicago Lectures in Mathematics Series

Geometry, Rigidity, and Group Actions

Author: Robert J Zimmer,Benson Farb,David Fisher
Publisher: University of Chicago Press
ISBN: 0226237893
Category: Mathematics
Page: 646
View: 9478

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The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others. The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.

Anschauliche Geometrie

Author: David Hilbert,Stefan Cohn-Vossen
Publisher: Springer-Verlag
ISBN: 3662366851
Category: Mathematics
Page: 312
View: 516

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Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

Lectures on Mathematics

Author: Felix Klein
Publisher: American Mathematical Soc.
ISBN: 0821827332
Category: Mathematics
Page: 109
View: 4933

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In the late summer of 1893, following the Congress of Mathematicians held in Chicago, Felix Klein gave two weeks of lectures on the current state of mathematics. Rather than offering a universal perspective, Klein presented his personal view of the most important topics of the time. It is remarkable how most of the topics continue to be important today. Originally published in 1893 and reissued by the AMS in 1911, we are pleased to bring this work into print once more with this new edition. Klein begins by highlighting the works of Clebsch and of Lie. In particular, he discusses Clebsch's work on Abelian functions and compares his approach to the theory with Riemann's more geometrical point of view. Klein devotes two lectures to Sophus Lie, focussing on his contributions to geometry, including sphere geometry and contact geometry. Klein's ability to connect different mathematical disciplines clearly comes through in his lectures on mathematical developments. For instance, he discusses recent progress in non-Euclidean geometry by emphasizing the connections to projective geometry and the role of transformation groups. In his descriptions of analytic function theory and of recent work in hyperelliptic and Abelian functions, Klein is guided by Riemann's geometric point of view. He discusses Galois theory and solutions of algebraic equations of degree five or higher by reducing them to normal forms that might be solved by non-algebraic means. Thus, as discovered by Hermite and Kronecker, the quintic can be solved "by elliptic functions". This also leads to Klein's well-known work connecting the quintic to the group of the icosahedron. Klein expounds on the roles of intuition and logical thinking in mathematics. He reflects on the influence of physics and the physical world on mathematics and, conversely, on the influence of mathematics on physics and the other natural sciences. The discussion is strikingly similar to today's discussions about ``physical mathematics''. There are a few other topics covered in the lectures which are somewhat removed from Klein's own work. For example, he discusses Hilbert's proof of the transcendence of certain types of numbers (including $\pi$ and $e$), which Klein finds much simpler than the methods used by Lindemann to show the transcendence of $\pi$. Also, Klein uses the example of quadratic forms (and forms of higher degree) to explain the need for a theory of ideals as developed by Kummer. Klein's look at mathematics at the end of the 19th Century remains compelling today, both as history and as mathematics. It is delightful and fascinating to observe from a one-hundred year retrospect, the musings of one of the masters of an earlier era.

Fuchsian Groups

Author: Svetlana Katok
Publisher: University of Chicago Press
ISBN: 9780226425825
Category: Mathematics
Page: 175
View: 7165

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This introductory text provides a thoroughly modern treatment of Fuchsian groups that addresses both the classical material and recent developments in the field. A basic example of lattices in semisimple groups, Fuchsian groups have extensive connections to the theory of a single complex variable, number theory, algebraic and differential geometry, topology, Lie theory, representation theory, and group theory.