Mathematical Surveys and Monographs Volume 215 On Groups of PL-homeomorphisms of the Real Line Robert Bieri Ralph Strebel American Mathematical Society https://doi.org/10.1090//surv/215 On Groups of PL-homeomorphisms of the Real Line Mathematical Surveys and Monographs Volume 215 On Groups of PL-homeomorphisms of the Real Line Robert Bieri Ralph Strebel American Mathematical Society Providence, Rhode Island EDITORIAL COMMITTEE Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein 2010 Mathematics Subject Classification. Primary 20-02, 20F05, 20E32, 20E36; Secondary 20F38. For additional information and updates on this book, visit www.ams.org/bookpages/surv-215 Library of Congress Cataloging-in-Publication Data Names: Bieri, Robert. | Strebel, Ralph, 1944- Title: On groups of PL-homeomorphisms of the real line / Robert Bieri, Ralph Strebel. Description: Providence, Rhode Island : American Mathematical Society, [2016] | Series: Mathe- matical surveys and monographs ; volume 215 | Includes bibliographical references and index. Identifiers: LCCN 2016025287 | ISBN 9781470429010 (alk. paper) Subjects: LCSH: Homeomorphisms. | Geometric group theory. | Numbers, Real. | Dimension theory (Algebra) | AMS: Group theory and generalizations – Research exposition (monographs, survey articles). msc | Group theory and generalizations – Structure and classification of infinite or finite groups – Simple groups. msc | Group theory and generalizations – Structure and classification of infinite or finite groups – Automorphisms of infinite groups. msc | Group theory and generalizations – Special aspects of infinite or finite groups – Generators, relations, and presentations. msc | Group theory and generalizations – Special aspects of infinite or finite groups – Other groups related to topology or analysis. msc Classification: LCC QA614 .B54 2016 | DDC 512/.2–dc23 LC record available at https://lccn. loc.gov/2016025287 Copying and reprinting. 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Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2016 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 Contents Preface vii Introduction 1 1. Background 1 2. Outline of our investigation 4 Chapter A. Construction of Finitary PL-homeomorphisms 13 3. Preliminaries 13 4. The basic result 15 5. Applications 17 Chapter B. Generating Sets 23 6. Necessary conditions for finite generation 23 7. Generators and relations for groups with supports in the line 23 8. Generators and relations for groups with supports in a half line 24 9. Generators for groups with supports in a compact interval 28 Chapter C. The Subgroup of Bounded Homeomorphisms B 43 10. Simplicity of the derived group of the subgroup B 43 11. Construction of homomorphisms into the slope group 45 12. Investigation of the abelianization of the subgroup B 49 Chapter D. Presentations 61 13. Presentations of groups with supports in the line 61 14. Presentations of groups with supports in a half line 71 15. Presentations of groups with supports in a compact interval 85 Chapter E. Isomorphisms and Automorphism Groups 89 16. General results 89 17. Isomorphisms of groups with non-cyclic slope groups 101 18. Isomorphisms of groups with cyclic slope groups 110 19. Automorphism groups of groups with cyclic slope groups 125 Notes 137 N1. Differences between memoir and monograph: summary 137 N2. Differences between memoir and monograph: details 138 N3. Related articles 145 v vi CONTENTS Bibliography 163 Index of Notation 167 Subject Index 171 Preface The main part of this monograph is a corrected and augmented version of the Bieri–Strebel memoir [BS85a]. The manuscript of that memoir was completed in the autumn of 1985; it was then typed and the typescript was distributed to a few mathematicians. At the beginning of 2014, I started to revise it, my original plan being to stay close to the original text. Little by little, I had to abandon this idea for reasons explained in Section N1 of the chapter called Notes. That chapter contains two further sections: Section N2 details the differences between the memoir and this monograph, and Section N3 surveys results obtained after 1985 and related to the topics of the memoir. At the end of 2014, a corrected version of the memoir was published in the repository arXiv as [BS14]. The present monograph is based on this electronic preprint, but the text has been reworked once more, the chapter Notes and the bibliography have been updated, and two indices have been added. Origin of the memoir In May 1984, Robert Bieri and I attended a meeting at the Mathematisches Forschungsinstitut Oberwolfach (Germany). There Ross Geoghegan introduced us to some recent discoveries of Matthew Brin and Craig Squier concerning groups of piecewise linear homeomorphisms of the real line; many of these findings were to be published later on in [BS85b]. These discoveries are a fascinating mixture of general results and explicit com- putations with specific groups. To describe them, I need a bit of notation. Let PLo(R) denote the group of all orientation preserving piecewise linear homeomor- phisms of the real line with only finitely many breaks. It is easy to see that this group is torsion-free and locally indicable. Brin and Squier establish that it satisfies no laws and that it contains no non-abelian free subgroups. The group PLo(R) has intriguing subgroups: first a finitely presented subgroup, discovered by Richard J. Thompson in about 1965, later rediscovered independently by Freyd and Heller (from a homotopy viewpoint) and by Dydak and Minc (from a shape theory viewpoint) (see [FH93]and[Dyd77]). The group has a very peculiar infinite presentation, namely xi (1) F = x0,x1,x2,...| xj = xj+1 for all indices i<j. (I use left action, as employed in the memoir [BS85a].) According to Thompson’s manuscript [Tho74], the derived group of F is a minimal normal subgroup of F ; it is actually simple, a fact discovered around 1969 by Freyd and Heller (see part (T3) of the Main Theorem in [FH93]), but known to Thompson, too. vii viii PREFACE Brin and Squier studied a second subgroup G of PLo(R). It consists of all PL- homomorphisms of R with supports contained in the interval [0, ∞[ , having only finitely many singularities, all at dyadic rationals, and with slopes that are powers of 2. The group G has the infinite presentation xi (2) x0,x1,x2,...| xj = x2j−i for all indices i<j, but it has also the finite presentation 2 2 4 x1x0 x x1x x (3) x0,x1, | x1 = 0 x1 and 0 x1 = 0 x1. The subgroups of PLo(R) exhibit further surprises. In [Tho74], R. J. Thomp- son represents the group F by PL-homeomorphisms with supports in the unit in- terval [0, 1]; as discovered by him and, independently, by Freyd and Heller, it can also be realized by PL-homeomorphisms with supports in the half line [0, ∞[or with supports in the real line.1 Layout of the memoir The groups F and G have explicit finite presentations with generating sets made up of concretely given PL-homeomorphisms. In addition, they have definitions that might be called local: the group F consists of all PL-homeomorphisms with supports in the unit interval I =[0, 1], slopes in the multiplicative group P generated by 2 and breaks in the Z[P ]-submodule Z[1/2] of the additive group of R, while G is made up of all PL-homeomorphisms with supports in the half line I =[0, ∞[, slopes in P = gp(2) and breaks in A = Z[1/2]. These local definitions of F and of G form the starting point of the Bieri– Strebel memoir [BS85a]. Similarly to what Brin and Squier propose on page 490 of [BS85b], the authors consider subgroups of PLo(R), depending on three parameters I, A and P ,whereI ⊆ R is a closed interval, P is a subgroup of the R× Z multiplicative group of the positive reals >0 and A is a [P ]-submodule of the additive group Radd. To every such triple they attach the subset G(I; A, P )={g ∈ PLo(R) | g has support in I,slopesinP ,breaksinA}. This subset is closed under composition of functions and under passage to the inverse2 and G(I; A, P ) equipped with this composition is a group which, by abuse of notation, will again be denoted by G(I; A, P ). In order to avoid trivialities, the authors require, in addition, that I have positive length, that P = {1} and that A = {0}. These non-triviality assumptions imply that A is a dense subgroup of R. Themes. The group F has many striking properties; the following five are chosen by Bieri and Strebel as themes of their memoir: (i) F is dense in the space of all orientation preserving homeomorphisms of the compact interval [0, 1],3 (ii) its derived group is simple, (iii) F is finitely generated, (iv) F admits a finite presentation, and (v) F is isomorphic to subgroups of PLo(R) with supports in [0, ∞[.