Algorithmic Problems in Groups and Semigroups

311 Pages·2000·20.018 MB·English

by  Jorge Almeida, Benjamin Steinberg (auth.), Jean-Camille Birget, Stuart Margolis, John Meakin, Mark Sapir (eds.)

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Algorithmic Problems
in Groups
and Semigroups
J.-C. Birget
s.
Margolis
J. Meakin
M. Sapir
Editors
Springer-Science+Business Media, LLC
Jean-Camille Birget Stuart Margolis
Faeulty of Computer Scienee Dept. of Mathematies & Computer Scienee
Dalhousie University Bar-Han University
Halifax, Nova Seotia 52900 Ramat Gan
Canada B3J 3K5 Israel
John Meakin Mark Sapir
Dept. of Mathematies & Statisties Dept. of Mathematies
University of Nebraska Vanderbilt University
Lineoln, NE 68588-0323 Nashville, TN 37240
USA USA
Ubrary of Congress Cataloging-in-Publication Data
Algorithmie problems in groups and semigroups / J.-c. Birget ... let al.], editors.
p. cm. - (Trends in Mathematics)
Includes bibliographical references.
ISBN 978-1-4612-7126-0 ISBN 978-1-4612-1388-8 (eBook)
DOI 10.10071978-1-4612-1388-8
I. Group theory. 2. Semigroups. 3. Algorithms. I. Birget, J.-C. (Jean-Camille) 11.
Series.
QAI74.2.A44 2000
512'.2-de21 99-052747
CIP
AMS Subject Classifications: OB25, 20-06, 20Mxx
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Birkhliuser Boston in 2000
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SPIN 10723749
ISBN 978-1-4612-7126-0
Reformatted from electronic files in It\Tp' by lohn Spiegelman, Philadelphia, PA.
9 8 7 6 5 4 3 2 1
Contents
Preface
Stuart Margolis, Mark Sapir, Jean-Camille Birget, and John Meakin vii
Invited Lecturers ix
Additional Lecturers x
Syntactic and Global Semigroup Theory: A Synthesis Approach
Jorge Almeida and Benjamin Steinberg 1
Semigroups with Central Idempotents
Karl Auinger 25
Algebraic Geometry over Groups
Gilbert Baumslag, Alexei Myasnikov, and Vladimir Remeslennikov 35
Aspects of the Theory of Free Groups
Katalin Bencsath, Benjamin Fine, Anthony M. Gaglione, Alexei G. Myasnikov,
Frank Roehl, Gerhard Rosenberger, and Dennis Spellman 51
Polynomial Isoperimetric Inequalities for Richard Thompson's Groups
F, T, and V
V.S. Guba 91
vi Contents
Ordered Monoids and .J- Trivial Monoids
Karsten Henckell and Jean-Eric Pin 121
A Remark on Finitely Generated Subgroups of Free Groups
S. V. Ivanov and P.E. Schupp 139
Homotopy Reduction Systems for Monoid Presentations II:
The Guba-Sapir Reduction and Homotopy Modules
YujiKobayashi 143
Algorithmic Problems for Finite Groups and Finite Semi groups
S.I. Kublanovskii 161
A Survey on the Computational Power of Some Classes
of Finite Monoid Presentations
Friedrich Otto 171
Rewriting Systems, Finiteness Conditions, and Associated Functions
SJ. Pride and Jing Wang 195
Multiparty Communication Complexity of Finite Monoids
Jean-Franfiois Raymond, Pascal Tesson, and Denis Therien 217
Presentations for Monoids, Their Maximal Subgroups,
and Schiitzenberger Groups
N. Ruskuc 235
On the Growth of Relatively Free Semigroups
L.M. Shneerson 251
When Can One Finite Monoid Simulate Another?
Howard Straubing 267
Computing Closures of Finitely Generated Subgroups of the Free Group
Pascal Wei! 289
Preface
This volume contains papers which are based primarily on talks given at an inter
national conference on Algorithmic Problems in Groups and Semigroups held at
the University of Nebraska-Lincoln from May ll-May 16, 1998. The conference
coincided with the Centennial Celebration of the Department of Mathematics and
Statistics at the University of Nebraska-Lincoln on the occasion of the one hun
dredth anniversary of the granting of the first Ph.D. by the department. Funding
was provided by the US National Science Foundation, the Department of Math
ematics and Statistics, and the College of Arts and Sciences at the University of
Nebraska-Lincoln, through the College's focus program in Discrete, Experimental
and Applied Mathematics.
The purpose of the conference was to bring together researchers with interests
in algorithmic problems in group theory, semigroup theory and computer science.
A particularly useful feature of this conference was that it provided a framework
for exchange of ideas between the research communities in semigroup theory
and group theory, and several of the papers collected here reflect this interac
tion of ideas.
The papers collected in this volume represent a cross section of some of the
results and ideas that were discussed in the conference. They reflect a synthesis
of overlapping ideas and techniques stimulated by problems concerning finite
monoids, finitely presented mono ids, finitely presented groups and free groups.
Several of the papers in this volume are either expository or have large expository
components, so the volume can be used both as an introduction to the subject of
algorithmic problems in groups and semigroups and as a reflection of the current
state of the subject.
The algorithmic theory of free groups is discussed in survey articles by Baum
slag/Myasnikov/Remeslennikov and Bencsath/Fine/Gaglione/Myasnikov/ Roehl/
viii Preface
Rosenberger/Spellman. The first of these papers describes the algebro-geometric
approach to free groups, which has proved to be very useful in the study of the
elementary theories of these groups and Tarski's problems. The second article
surveys classical and modern aspects of the theory of free groups. These two papers
complement each other in a natural way. Two other papers in this volume are also
devoted to free groups and their subgroups. Ivanov/Schupp discusses amalgamated
products of free groups, and Weil surveys several new and old results related to
different profinite topologies on free groups.
Weil's paper is really in the common territory of the theory of free groups and
the global theory of finite semigroups, which studies pseudo-varieties of finite
semigroups, their decompositions and Rhodes complexity of finite semigroups.
The interplay of these two theories has proved to be very useful for both of them.
Rhodes complexity is one of the main topics of the papers by Almeida/Steinberg,
Auinger, and HenckelllPin.
One of the central problems concerning Rhodes complexity of finite semigroups
is whether the complexity of a finite semigroup is decidable. Recent results by
Kublanovskii and others show that several natural problems about finite semi
groups, and even finite simple semigroups, are undecidable. Kublanovskii surveys
recent results in this direction.
There exists a fruitful connection between the algebraic theory of finite monoids
and the theory of communication complexity. Results in this direction are surveyed
in the paper by Raymond/Tesson/Therien. The paper by Straubing explores the
connection between circuit complexity and simulation by finite monoids.
Several papers discuss algorithmic properties of finite semigroup presenta
tions and Thue (string rewriting) systems. String rewriting techniques are used
in Otto's paper to study decidabilty and complexity questions about finitely pre
sented monoids, while homological techniques and ideas from low dimensional
homotopy are used in the papers by Pride/Wang and Kobayashi to study finite
ness properties and rewrite systems for finitely presented monoids. Computational
aspects of finitely presented monoids and their subgroups are studied by Ruskuc.
The study of asymptotic functions of groups and semigroups provides another
area where connections between semigroups and groups have been mutually ben
eficial. Shneerson surveys recent results about growth functions of finitely gener
ated semigroups. Guba surveys the diagram group approach in studying the Dehn
functions of R. Thompson's groups. Both papers contain new results (with proofs)
which have never been published before as well as surveys of published results
and methods.
Stuart Margolis
Mark Sapir
Jean-Camille Birget
John Meakin
August 1999
Invited Lecturers
Jorge Almeida (Porto, Portugal)
Gilbert Baumslag (City College, New York)
Robert Gilman (Stevens Institute of Technology, New Jersey)
Victor Guba (Vologda, Russia)
Derek Holt (Warwick, England)
Sergei Ivanov (Urbana-Champaign, Illinois)
Olga Kharlampovich (McGill, Canada)
Yuji Kobayashi (Toho University, Japan)
Stanislav Kublanovskii (St. Petersburg, Russia)
Stuart Margolis (Bar Ilan, Israel)
Alexei Myasnikov (City College, New York)
Friedrich Otto (Kassel, Germany),
A. Yu. Ol'shanskii (Moscow, Russia)
Jean-Eric Pin (Paris, France)
Steve Pride (Glasgow, Scotland)
John Rhodes (Berkeley, California)
Nik Ruskuc (St Andrews, Scotland)
Mark Sapir (Vanderbilt, Tennessee)
Paul Schupp (Urbana-Champaign, Illinois)
Lev Shneerson (New York)
Ben Steinberg (Berkeley, California)
Howard Straubing (Boston College, Massachusetts)
Denis Therien (McGill, Canada)
Mikhail Volkov (Ural State University, Russia)
Pascal Weil (Paris, France)
Additional Lecturers
Juan Alonso (Stockholm, Sweden)
Karl Auinger (Vienna, Austria)
Hayrullah Ayik (St Andrews, Scotland)
Colin Campbell (St Andrews, Scotland)
David Cruichshank (Glasgow, Scotland)
Victor Fernandes (Lisbon, Portugal)
Anthony Gaglione (US Naval Academy, Annapolis, Maryland)
Pedro Garda-Sanchez (Granada, Spain)
Karsten Henckell (New College, Florida)
Susan Hermiller (New Mexico State, New Mexico)
Tanya Jajcayova (Bratislava, Slovakia)
Peter Jones (Marquette University, Wisconsin)
Sylvain Lombardy (ENS Paris, France)
Jon McCammond (Texas A & M, Texas)
James Renshaw (Southampton, England)
Robert Ruyle (Lincoln, Nebraska)
Olga Sapir (Vanderbilt, Tennessee)
Steve Seif (Louisville, Kentucky)
Pedro Silva (Porto, Portugal)
Jean-Claude Spehner (Universite Haute-Alsace, France)
Joseph Stephen (Northern Illinois University, Illinois)
Rick Thomas (Leicester, England)
Jing Wang (Glasgow, Scotland)
Akihiro Yamamura (Tokyo, Japan)