Title Lectures on Profinite Topics in Group Theory

Published Cambridge : Cambridge University Press, 2011

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Description 1 online resource (160 pages) Series London Mathematical Society Student Texts, 77 ; v. 77 London Mathematical Society Student Texts, 77 Contents Cover; Title; Copyright; Contents; Preface; Editor's introduction; I An introduction to compact p-adic Lie groups; 1 Introduction; 2 From finite p-groups to compact p-adic Lie groups; 2.1 Nilpotent groups; 2.2 Finite p-groups; 2.3 Lie rings; 2.4 Applying Lie methods to groups; 2.5 Absolute values; 2.6 p-adic numbers; 2.7 p-adic integers; 2.8 Preview: p-adic analytic pro-p groups; 3 Basic notions and facts from point-set topology; 4 First series of exercises; 5 Powerful groups, profinite groups and pro-p groups; 5.1 Powerful finite p-groups; 5.2 Profinite groups as Galois groups 5.3 Profinite groups as inverse limits5.4 Profinite groups as profinite completions; 5.5 Profinite groups as topological groups; 5.6 Pro-p groups; 5.7 Powerful pro-p groups; 5.8 Pro-p groups of finite rank -- summary ofcharacterisations; 6 Second series of exercises; 7 Uniformly powerful pro-p groups and Zp-Lie lattices; 7.1 Uniformly powerful pro-p groups; 7.2 Associated additive structure; 7.3 Associated Lie structure; 7.4 The Hausdorff formula; 7.5 Applying the Hausdorff formula; 8 The group GLd(Zp), just-infinite pro-p groups and the Lie correspondence for saturable pro-p groups 8.1 The group GLd(Zp) -- an example8.2 Just-infinite pro-p groups; 8.3 Potent filtrations and saturable pro-p groups; 8.4 Lie correspondence; 9 Third series of exercises; 10 Representations of compact p-adic Lie groups; 10.1 Representation growth and Kirillov's orbit method; 10.2 The orbit method for saturable pro-p groups; 10.3 An application of the orbit method; References for Chapter I; II Strong approximation methods; 1 Introduction; 2 Algebraic groups; 2.1 The Zariski topology on Kn; 2.2 Linear algebraic groups as closed subgroups of GLn(K); Basic examples Basic properties of Algebraic groupsFields of definition and restriction of scalars; The Lie algebra of G; Connection with Lie algebras of locally compact topological groups; 2.3 Semisimple algebraic groups: the classification ofsimply connected algebraic groups over K; 2.4 Reductive groups; 2.5 Chevalley groups; 3 Arithmetic groups and the congruence topology; 3.1 Rings of algebraic integers in number fields; 3.2 The congruence topology on GLn(k) and GLn(O); Valuations of k; 3.3 Arithmetic groups; 4 The strong approximation theorem; 4.1 An aside: Serre's conjecture; 5 Lubotzky's alternative 6 Applications of Lubotzky's alternative6.1 The finite simple groups of Lie type; 6.2 Refinements; 6.3 Normal subgroups of linear groups; 6.4 Representations, sieves and expanders; 7 The Nori -- Weisfeiler theorem; 7.1 Unipotently generated subgroups of algebraic groups over finite fields; 8 Exercises; References for Chapter II; III A newcomer's guideto zeta functions of groups and rings; 1 Introduction; 1.1 Zeta functions of groups; 1.2 Zeta functions of rings; 1.3 Linearisation; 1.4 Organisation of the chapter; 2 Local and global zeta functions; 2.1 Rationality and variation with the prime Summary An introduction to three key aspects of current research in infinite group theory, suitable for graduate students Notes 2.2 Flag varieties and Coxeter groups Bibliography Includes bibliographical references and index Notes Print version record Subject Profinite groups. Group theory. MATHEMATICS -- Group Theory. Teoría de grupos Grupos profinitos Group theory Profinite groups Gruppentheorie Proendliche Gruppe Form Electronic book Author Nikolov, Nikolay Voll, Christopher Segal, Dan ISBN 9781139117593 1139117599 9781139128254 1139128256 9781139115421 1139115421 9780511793837 0511793839

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