Table Of ContentAdvanced Courses in Mathematics - CRM Barcelona
Javier Aramayona
Volker Diekert
Christopher J. Leininger
Pedro V. Silva
Armin Weiß
Algorithmic
and Geometric Topics
Around Free Groups
and Automorphisms
Advanced Courses in Mathematics
CRM Barcelona
Centre de Recerca Matemàtica
Managing Editor:
Enric Ventura
More information about this series at http://www.springer.com/series/5038
Javier Aramayona • Volker Diekert
Christopher J. Leininger • Pedro V. Silva
Armin Weiß
Algorithmic and Geometric
Topics Around Free Groups
and Automorphisms
Editors for this volume:
Juan González-Meneses, Universidad de Sevilla
Martin Lustig, Université d’Aix-Marseille
Enric Ventura, Universitat Politècnica de Catalunya
Javier Aramayona Volker Diekert
Universidad Autónoma de Madrid - ICMAT Institut für Formale Methoden
Madrid, Spain der Informatik
Universität Stuttgart
Christopher J. Leininger Stuttgart, Germany
Department of Mathematics
University of Illinois Urbana-Champaign Pedro V. Silva
Urbana, IL, USA Centro de Matemática
Universidade do Porto
Armin Weiß Porto, Portugal
Institut für Formale Methoden der Informatik
Universität Stuttgart
Stuttgart, Germany
ISSN 2297-0304 ISSN2297- 0312 (electr onic)
Advanced Courses in Mathematics - CRM Barcelona
ISBN 978-3-319-60939-3 ISBN 978-3-319-60940-9 (eBook)
DOI 10.1007/978-3-319-60940-9
Library of Congress Control Number: 2017955841
Mathematics Subject Classification (2010): 20E05, 20E08, 20E36, 20M35, 20F67, 68Q45
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Foreword
The Research Programme on Automorphisms of Free Groups: Algorithms, Ge-
ometry and Dynamics was organized and took place at the Centre de Recerca
Matema`tica (CRM), Bellaterra, Barcelona, from September to December 2012. It
wascoordinatedbyJuanGonza´lez-Meneses(UniversidaddeSevilla,Spain),Martin
Lustig (Universit´e Aix-Marseille III, Marseille, France), Alexandra Pettet (Uni-
versity of British Columbia, Vancouver, Canada), and Enric Ventura (Universitat
Polit`ecnica de Catalunya, Barcelona, Catalonia).
Duringtheseintensefourmonthsseveralscientificeventstookplace,including
the Summer School on Automorphisms of Free Groups (held from September 25th
to 29th, 2012), the international Conference on Automorphisms of Free Groups
(heldfromNovember12thto16th,2012),twoworkshops,andaweeklyseminar,all
ofthemwiththeactiveparticipationofmanyvisitorsinvitedtoattendfromseveral
countries abroad. Of course, in addition, all participants had numerous occasions
for informal but fruitful conversations among themselves, discussing mathematical
ideas which, in many cases, gave rise to new and interesting results, published
later in regular mathematical research journals. Altogether, it was a very exciting
and stimulating experience, which contributed to the development of our research
area, and gave extra opportunities to researchers and young students to work and
interact with each other, in a very dynamic and productive research atmosphere.
In the present volume of the series Advanced Courses in Mathematics CRM
Barcelona,wepresentanextendedandelaboratedversionofthenotescorresponding
tothethreecoursestaughtattheSummerSchoolonAutomorphismsofFreeGroups,
by internationally recognized experts in the area.
In the first chapter, entitled “An Automata-Theoretic Approach to the Study
of Fixed Points of Endomorphisms”, professor Pedro Silva presents the basic
definitions and results on automata theory, and an interesting review of their
important role in several parts of group theory. Particularly interesting is the
relationbetweenautomataandendomorphismsofvirtuallyfreegroups;thechapter
compiles a brief history of the study of (finite and infinite) fixed points of group
endomorphisms, and discusses the case of virtually free groups with the help of
automata.
The second chapter, “Context-Free Groups and Bass–Serre Theory”, by pro-
fessors Volker Diekert and Armin Weiß, also concentrates on virtually free groups,
v
vi Foreword
but from a more language theoretic point of view: by the classical Muller–Schupp
theorem, the family of (finitely generated) virtually free groups happens to be the
same as that of context-free groups, and this allows the authors to investigate
them with techniques from language theory (automata, regular and context-free
languages, rewriting systems, etc.). From this chapter we would like to point to an
introduction to Bass–Serre theory using rewriting systems, and a self-contained
access to the Muller–Schupp theorem without using Stallings’ structure theorem
or the accessibility result by Dunwoody.
Finally, the third and last chapter is entitled “Hyperbolic Structures on Sur-
faces and Geodesic Currents”. Switchingintoamore geometric territory, professors
Javier Aramayona and Christopher J. Leininger give an account of Bonahon’s
descriptionofThurston’scompactificationofTeichmu¨llerspaceintermsofgeodesic
currents on surfaces: it is explained why a surface equipped with a complete hy-
perbolic structure is isometric to the quotient of H2 by a Fuchsian group; then,
after reviewing some basic features of Teichmu¨ller spaces and measured geodesic
laminations, and after some words about the “classical” construction of Thurston’s
compactification, the authors introduce geodesic currents, and develop Bonahon’s
interpretation of the compactification of Teichmu¨ller space, giving then some gen-
eralizations of the notion of geodesic currents to other settings (such as negatively
curved metrics on surfaces, flat metrics on surfaces, and free groups).
We would like to express our gratitude to CRM for hosting and supporting
our research programme. We also convey our warm thanks to the CRM Director,
Joaquim Bruna, and to the secretaries and staff of the center for providing great
facilities, and a very pleasant working environment. Our gratitude also to the
lecturers and authors of these notes, Pedro Silva, Volker Diekert, Armin Weiß,
Javier Aramayona, and Christopher J. Leininger for the enthusiasm and energy
showed during the courses, for their careful preparation of these notes, and for
their patience in the process of publication. Finally, our special thanks also to
those who attended the courses, young and senior, for their interest, their active
participation, and their enthusiasm towards mathematics.
Sevilla, Juan Gonza´lez-Meneses,
Marseille, Martin Lustig,
Barcelona, Enric Ventura.
April 2017.
Contents
1 An Automata-Theoretic Approach to the Study of Fixed Points of
Endomorphisms 1
By Pedro Silva
1.1 Languages and Automata . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Words and Languages . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Transducers . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Rewriting Systems . . . . . . . . . . . . . . . . . . . . . . . 8
1.2 Automata in Group Theory . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Virtually Free Groups . . . . . . . . . . . . . . . . . . . . . 10
1.2.3 Hyperbolic Groups . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.4 Automatic Groups . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.5 Self-Similar Groups. . . . . . . . . . . . . . . . . . . . . . . 15
1.3 Fixed Points of Endomorphisms . . . . . . . . . . . . . . . . . . . . 17
1.3.1 A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . 17
1.3.2 Fixed Points of Transductions . . . . . . . . . . . . . . . . . 18
1.3.3 Virtually Free Group Endomorphisms . . . . . . . . . . . . 21
1.4 Fixed Points in the Boundary . . . . . . . . . . . . . . . . . . . . . 23
1.4.1 A Brief Introduction . . . . . . . . . . . . . . . . . . . . . . 23
1.4.2 A Model for the Boundary of Virtually Free Groups . . . . 24
1.4.3 Uniformly Continuous Endomorphisms. . . . . . . . . . . . 28
1.4.4 Fixed Points in the Boundary of Virtually Free Groups . . 29
1.4.5 Classification of the Infinite Fixed Points . . . . . . . . . . 34
1.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2 Context-Free Groups and Bass–Serre Theory 43
By Volker Diekert and Armin Weiß
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
vii
viii Contents
2.2.1 Rewriting Systems . . . . . . . . . . . . . . . . . . . . . . . 45
2.2.2 Presentations of Monoids and Groups . . . . . . . . . . . . 46
2.2.3 Preliminaries on Graphs . . . . . . . . . . . . . . . . . . . . 47
2.3 Formal Language Theory for Groups . . . . . . . . . . . . . . . . . 50
2.3.1 Regular Languages . . . . . . . . . . . . . . . . . . . . . . . 50
2.3.2 The Chomsky Hierarchy . . . . . . . . . . . . . . . . . . . . 53
2.3.3 Context-Free Languages . . . . . . . . . . . . . . . . . . . . 55
2.4 Bass–Serre Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
2.4.1 Britton Reductions over Graphs of Groups . . . . . . . . . 67
2.4.2 Bass–Serre Tree. . . . . . . . . . . . . . . . . . . . . . . . . 69
2.4.3 Groups Acting on Trees . . . . . . . . . . . . . . . . . . . . 69
2.4.4 Finite Vertex Groups. . . . . . . . . . . . . . . . . . . . . . 72
2.4.5 Embedding into Semidirect Products . . . . . . . . . . . . . 73
2.5 Pregroups and Geodesic Rewriting Systems . . . . . . . . . . . . . 75
2.5.1 Finite Graphs of Groups and Pregroups . . . . . . . . . . . 78
2.6 Graphs and Treewidth . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.6.1 Cayley Graphs . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.6.2 Quasi-Isometries and Treewidth. . . . . . . . . . . . . . . . 89
2.7 Cuts and Structure Trees . . . . . . . . . . . . . . . . . . . . . . . 90
2.7.1 Cuts in Graphs of Finite Treewidth . . . . . . . . . . . . . 91
2.7.2 Optimally Nested Cuts . . . . . . . . . . . . . . . . . . . . 93
2.7.3 The Structure Tree . . . . . . . . . . . . . . . . . . . . . . . 98
2.7.4 Actions on the Structure Tree . . . . . . . . . . . . . . . . . 99
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3 Hyperbolic Structures on Surfaces and Geodesic Currents 111
By Javier Aramayona and Christopher J. Leininger
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
3.2 Hyperbolic Structures on Surfaces . . . . . . . . . . . . . . . . . . 112
3.3 Teichmu¨ller Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
3.3.1 Length Functions . . . . . . . . . . . . . . . . . . . . . . . . 116
3.3.2 Fenchel–Nielsen Coordinates . . . . . . . . . . . . . . . . . 117
3.3.3 Measured Laminations and Thurston’s Compactification of
T(S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4 Geodesic Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.4.1 Measured Laminations as Measures on G(S˜) . . . . . . . . 122
3.4.2 Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.4.3 Alternative Definition of Geodesic Currents . . . . . . . . . 123
3.4.4 The Flow-Box Topology on Curr(S) . . . . . . . . . . . . . 126
3.4.5 Intersection Number Between Geodesic Currents . . . . . . 127
3.4.6 Projective Currents . . . . . . . . . . . . . . . . . . . . . . 129
3.4.7 Determining Currents from Intersection Numbers . . . . . . 131
Contents ix
3.4.8 Liouville Currents . . . . . . . . . . . . . . . . . . . . . . . 133
3.4.9 Teichmu¨ller Space . . . . . . . . . . . . . . . . . . . . . . . 134
3.4.10 Thurston’s Compactification . . . . . . . . . . . . . . . . . 135
3.5 Geodesic Currents in Other Settings . . . . . . . . . . . . . . . . . 136
3.5.1 Liouville Measures for Riemannian Metrics . . . . . . . . . 137
3.5.2 From Liouville Measure to Liouville Current . . . . . . . . 140
3.5.3 Mo¨bius Currents . . . . . . . . . . . . . . . . . . . . . . . . 141
3.5.4 Quadratic Differentials . . . . . . . . . . . . . . . . . . . . . 143
3.5.5 Free Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
