Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture
This monograph presents some cornerstone results in the study of sofic and hyperlinear groups and the closely related Connes' embedding conjecture. These notions, as well as the proofs of many results, are presented in the framework of model theory for me
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Valerio Capraro Martino Lupini
Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and NY Catharina Stroppel, Bonn Anna Wienhard, Heidelberg
2136
More information about this series at http://www.springer.com/series/304
Valerio Capraro • Martino Lupini
Introduction to Sofic and Hyperlinear Groups and Connes’ Embedding Conjecture With an Appendix by Vladimir Pestov
123
Valerio Capraro Center for Mathematics and Computer Science (CWI) Amsterdam The Netherlands
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-19332-8 DOI 10.1007/978-3-319-19333-5
Martino Lupini Department of Mathematics California Institute of Technology Pasadena CA, USA
ISSN 1617-9692
(electronic)
ISBN 978-3-319-19333-5
(eBook)
Library of Congress Control Number: 2015945815 Mathematics Subject Classification (2010): 20F65, 20F69, 03C20, 03C98, 46L10, 46M07 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)
Preface
Analogy is one of the most effective techniques of human reasoning: When we face new problems, we compare them with simpler and already known ones, in the attempt to use what we know about the latter ones to solve the former ones. This strategy is particularly common in Mathematics, which offers several examples of abstract and seemingly intractable objects: Subsets of the plane can be enormously complicated but, as soon as they can be approximated by rectangles, then they can be measured; Uniformly finite
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