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Topological Methods in Group Theory is about the interplay between algebraic topology and the theory of infinite discrete groups. The author has kept three kinds of readers in mind: graduate students who have had an introductory course in algebraic topology and who need a bridge from common knowledge to the current research literature in geometric, combinatorial and homological group theory; group theorists who would like to know more about the topological side of their subject but who have been too long away from topology; and manifold topologists, both high- and low-dimensional, since the book contains much basic material on proper homotopy and locally finite homology not easily found elsewhere. The book focuses on two main themes: 1. Topological Finiteness Properties of groups (generalizing the classical notions of "finitely generated" and "finitely presented"); 2. Asymptotic Aspects of Infinite Groups (generalizing the classical notion of "the number of ends of a group"). Illustrative examples treated in some detail include: Bass-Serre theory, Coxeter groups, Thompson groups, Whitehead's contractible 3-manifold, Davis's exotic contractible manifolds in dimensions greater than three, the Bestvina-Brady Theorem, and the Bieri-Neumann-Strebel invariant. The book also includes a highly geometrical treatment of Poincaré duality (via cells and dual cells) to bring out the topological meaning of Poincaré duality groups. To keep the length reasonable and the focus clear, it is assumed that the reader knows or can easily learn the necessary algebra (which is clearly summarized) but wants to see the topology done in detail. Apart from the introductory material, most of the mathematics presented here has not appeared in book form before.

Group theory. --- Discrete groups. --- Algebraic topology. --- Topological manifolds. --- Groups, Theory of --- Substitutions (Mathematics) --- Algebra --- Manifolds (Mathematics) --- Topology --- Groups, Discrete --- Infinite groups --- Topological Groups. --- Topology. --- Topological Groups, Lie Groups. --- Group Theory and Generalizations. --- Analysis situs --- Position analysis --- Rubber-sheet geometry --- Geometry --- Polyhedra --- Set theory --- Algebras, Linear --- Groups, Topological --- Continuous groups --- Topological groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups

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Group theory --- Topological groups. Lie groups --- Topology --- topologie (wiskunde) --- wiskunde --- topologie

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The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail, while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book -- Discrete and Continuous -- be given equal attention. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Mathematics. --- Mathematics, general. --- Mathématiques --- Proof theory. --- Mathematics --- Logic, Symbolic and mathematical --- Study and teaching. --- Electronic books. -- local. --- Mathematics -- Study and teaching. --- Mathematics teachers -- Training of. --- Proof theory --- Mathematical Theory --- Physical Sciences & Mathematics --- Study and teaching --- Mathematics teachers --- Training of. --- Teacher training --- Algebra. --- Mathematical analysis. --- Analysis (Mathematics). --- Mathematical logic. --- Mathematical Logic and Foundations. --- Analysis. --- Algebra of logic --- Logic, Universal --- Mathematical logic --- Symbolic and mathematical logic --- Symbolic logic --- Algebra, Abstract --- Metamathematics --- Set theory --- Syllogism --- 517.1 Mathematical analysis --- Mathematical analysis --- Math --- Science --- Logic, Symbolic and mathematical. --- Global analysis (Mathematics). --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic

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The Art of Proof is designed for a one-semester or two-quarter course. A typical student will have studied calculus (perhaps also linear algebra) with reasonable success. With an artful mixture of chatty style and interesting examples, the student's previous intuitive knowledge is placed on solid intellectual ground. The topics covered include: integers, induction, algorithms, real numbers, rational numbers, modular arithmetic, limits, and uncountable sets. Methods, such as axiom, theorem and proof, are taught while discussing the mathematics rather than in abstract isolation. Some of the proofs are presented in detail, while others (some with hints) may be assigned to the student or presented by the instructor. The authors recommend that the two parts of the book -- Discrete and Continuous -- be given equal attention. The book ends with short essays on further topics suitable for seminar-style presentation by small teams of students, either in class or in a mathematics club setting. These include: continuity, cryptography, groups, complex numbers, ordinal number, and generating functions.

Mathematical logic --- Algebra --- Mathematical analysis --- Mathematics --- algebra --- analyse (wiskunde) --- wiskunde --- logica