Geometry and Topology Down Under
A Conference in Honour of Hyam Rubinstein 11–22 July 2011
The University of Melbourne, Parkville, Australia
Craig D. Hodgson William H. Jaco
Martin G. Scharlemann Stephan Tillmann
Editors
Geometry and Topology Down Under
A Conference in Honour of Hyam Rubinstein 11–22 July 2011
The University of Melbourne, Parkville, Australia
Craig D. Hodgson William H. Jaco
Martin G. Scharlemann Stephan Tillmann
Editors
597
Geometry and Topology Down Under
A Conference in Honour of Hyam Rubinstein 11–22 July 2011
The University of Melbourne, Parkville, Australia
Craig D. Hodgson William H. Jaco
Martin G. Scharlemann Stephan Tillmann
Editors
EDITORIAL COMMITTEE
2010 Mathematics Subject Classification. Primary 57M25, 57M27,
57M50, 57N10, 57Q15, 57Q45, 20F65, 20F67, 53A10, 53C43.
Library of Congress Cataloging-in-Publication Data
Geometry and topology down under: a conference in honour of Hyam
Rubinstein, July 11–22, 2011, The University of Melbourne,
Parkville, Australia / Craig D. Hodgson, William H. Jaco, Martin G.
Scharlemann, Stephan Tillmann, editors.
pages cm – (Contemporary mathematics ; volume 597) Includes
bibliographical references. ISBN 978-0-8218-8480-5 (alk. paper) 1.
Low-dimensional topology–Congresses. 2. Three-manifolds
(Topology)–Congresses.
I. Rubinstein, Hyam, 1948– honouree. II. Hodgson, Craig David,
editor of compilation. III. Jaco, William H., 1940– editor of
compilation. IV. Scharlemann, Martin G., 1948– editor of
compilation. V. Tillmann, Stephan, editor of compilation.
QA612.14.G455 2013 2013012326 516–dc23 Contemporary Mathematics
ISSN: 0271-4132 (print); ISSN: 1098-3627 (online)
DOI: http://dx.doi.org/10.1090/conm/597
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Contents
List of Participants xv
Survey and Expository Papers
What is an Almost Normal Surface? Joel Hass 1
The Ergodic Theory of Hyperbolic Groups Danny Calegari 15
Mapping Class Groups of 3–Manifolds, Then and Now Sungbok Hong and
Darryl McCullough 53
Stacks of Hyperbolic Spaces and Ends of 3-Manifolds B. H. Bowditch
65
Harmonic Maps and Integrable Systems Emma Carberry 139
Some of Hyam’s Favourite Problems Hyam Rubinstein 165
Research Papers
Almost Normal Surfaces with Boundary David Bachman, Ryan
Derby-Talbot, and Eric Sedgwick 177
Computational Topology with Regina: Algorithms, Heuristics and
Implementations
Benjamin A. Burton 195
Left-Orderability and Exceptional Dehn Surgery on Two-Bridge Knots
Adam Clay and Masakazu Teragaito 225
v
Networking Seifert Surgeries on Knots IV: Seiferters and Branched
Coverings Arnaud Deruelle, Mario Eudave-Munoz, Katura
Miyazaki,
and Kimihiko Motegi 235
The Groups of Fibred 2–Knots Jonathan A. Hillman 281
On the Number of Hyperbolic 3–Manifolds of a Given Volume Craig
Hodgson and Hidetoshi Masai 295
Seifert Fibered Surgery and Rasmussen Invariant Kazuhiro Ichihara
and In Dae Jong 321
Existence of Spherical Angle Structures on 3–Manifolds Feng Luo
337
3–Manifolds with Heegaard Splittings of Distance Two J. Hyam
Rubinstein and Abigail Thompson 341
Generating the Genus g + 1 Goeritz Group of a Genus g Handlebody
Martin Scharlemann 347
Preface
In July 2011, a two-week event, now known as the ‘Hyamfest’, was
held at the University of Melbourne. It consisted of a workshop and
a conference, both of which covered a broad range of topics in
Geometry and Topology, including hyperbolic geometry, symplectic
geometry and geometric topology.
These proceedings mirror the spirit of the event: They include
research arti- cles, expository articles and a set of Hyam
Rubinstein’s favourite problems, again covering a broad range of
topics. The editors would like to thank the authors for the work
they have put into their contributions, and the referees for their
commit- ment and efforts in their private task. The editors thank
Christine Thivierge for her assistance in preparing this
volume.
The workshop would not have been possible without the lecturers who
put a lot of energy into preparing and delivering three inspiring
lecture series, and their assistants who prepared problem sets and
ran discussion sessions. The high standard of the talks at the
conference contributed greatly to its success.
The event was sponsored by the Australian Mathematical Sciences
Institute, the Australian Mathematical Society, the Clay
Mathematics Institute and the Na- tional Science Foundation. The
Department of Mathematics and Statistics at the University of
Melbourne provided a wonderful conference environment, and staff at
the Institute and the Department provided invaluable help and
support.
The Editors March 2013
The Hyamfest
The conference and workshop Geometry & Topology Down Under
consisted of two exciting weeks of lectures and research talks in
the Department of Mathemat- ics and Statistics at the University of
Melbourne. The event brought together an impressive line-up of
guests from the United States, Europe and Asia, and was attended by
115 students and researchers. It attracted experts and emerging re-
searchers who reported on recent results and explored future
directions in Geometry and Topology. The conference was held in
honour of Hyam Rubinstein and cele- brated his contributions to
topology and his long-standing role as an advocate for the
mathematical sciences.
The workshop (11-15 July) and conference (18-22 July) covered a
broad range of topics in Geometry and Topology, including
hyperbolic and symplectic geometry, Heegaard splittings and
triangulations of 3-manifolds, and recent advances and applications
in the study of graph manifolds. The organisers interpreted the
topic of the workshop and conference broadly, so that the meeting
had appeal to group theorists, analysts, differential geometers and
low-dimensional topologists. The event was designed so that it was
beneficial not only to the experts in the field but also to early
career researchers and graduate students.
In the first week, short courses were given by Danny Calegari on
Ergodic Theory of Groups, Walter Neumann on Invariants of
hyperbolic 3–manifolds and Leonid Polterovich on Function theory on
symplectic manifolds. The short courses intro- duced honours and
postgraduate students, as well as early-career or established
researchers, to a broad range of methods and results. Each lecturer
gave a 75- minute lecture each day. Discussion sessions, which were
led by vibrant, early- career researchers, were held each
afternoon. The lectures were of exceptionally high standard, and
special notes and exercises were designed for the participants. All
lectures have been recorded and are, in addition to a wealth of
other material, available on the conference website:
www.ms.unimelb.edu.au/∼hyamfest.
The conference in the second week featured a line-up of 23
international experts who reported on a variety of new results. For
instance, Ian Agol gave a proof of Simon’s conjecture, David Gabai
reported on recent progress on the topology of ending lamination
space, Walter Neumann talked about a new geometric decom- position
for complex surface singularities, Yi Ni showed that Khovanov
homology with an extra module structure detects unlinks, and Gang
Tian described a new symplectic curvature flow. Moreover, each day
featured a “What is. . . ?” talk in the spirit of the Notices of
the AMS just before lunch. These talks were positively received by
both junior and senior researchers.
ix
x THE HYAMFEST
In conjunction with the conference, a free public lecture was given
by Danny Calegari on 19 July. The public lecture attracted media
attention, and many mem- bers of the public attended the
lecture.
The organisers were thrilled by the geographical distribution of
the 115 regis- tered participants, more than half of whom travelled
to Melbourne from overseas. There were 55 participants from
Australia, 36 from the USA, 12 from Japan, and the remaining ones
from Canada, China, France, Hungary, Israel, Korea, Mexico,
Singapore and the UK. Moreover, 26 of the 55 Australian
participants travelled from interstate.
The organisers feel that the event has helped to develop and
strengthen collab- orations between different research groups
within Australia and between groups in Australia and overseas, and
to inspire young scientists, graduate and undergraduate students to
engage in exploring the many exciting research problems in this
area of mathematics.
The organisers are grateful for generous funding by the Australian
Mathemati- cal Sciences Institute (AMSI), the Australian
Mathematical Society (AustMS), the Clay Mathematics Institute and
the National Science Foundation (NSF), which made this exciting
event possible. The organisers thank the staff at AMSI for their
help and support, and the Department of Mathematics and Statistics
at the Uni- versity of Melbourne for hosting this event and
providing a wonderful conference environment.
Organising Committee
James Carlson (Clay Mathematics Institute) Loretta Bartolini
(Oklahoma State University) Danny Calegari (California Institute of
Technology) Craig Hodgson (University of Melbourne) William Jaco
(Oklahoma State University) Amnon Neeman (Australian National
University) Paul Norbury (University of Melbourne) Arun Ram
(University of Melbourne) Stephan Tillmann (University of
Queensland) Penny Wightwick (University of Melbourne) Nick Wormald
(University of Waterloo)
Courses at the Hyamfest
Lecturer: Danny Calegari Assistant: Alden Walker
An introduction to the use of dynamical and probabilistic methods
in geo- metric group theory, especially as applied to hyperbolic
groups. I hope to discuss (central) limit theorems for random
geodesics and random walks, behaviour of characteristic functions
(e.g. (stable) commutator length, w- length, etc.) under random
homomorphisms, and a few other topics if time permits.
Invariants of Hyperbolic 3-Manifolds
Lecturer: Walter Neumann Assistant: Christian Zickert
This short course will concentrate on number-theoretic invariants
and num- ber theoretic methods in the study of 3-manifolds. The
first lecture will be a brief introduction to algebraic number
theory, followed by four lectures on 3-manifolds, concentrating
mostly on hyperbolic 3-manifolds.
Function Theory on Symplectic Manifolds
Lecturer: Leonid Polterovich Assistant: Daniel Rosen
Function spaces associated to a symplectic manifold exhibit
unexpected properties and interesting structures, giving rise to an
alternative intuition and new tools in symplectic topology. These
phenomena are detected by modern symplectic methods such as Floer
theory and are closely related to algebraic and geometric
properties of groups of Hamiltonian diffeomor- phisms. I shall
discuss these developments, their applications as well as links to
other areas such as group quasi-morphisms and quantum-classical
correspondence. All necessary symplectic preliminaries will be
explained.
Notes, problem sets and video recordings of the lectures are
available at:
http://www.ms.unimelb.edu.au/∼hyamfest/workshop.php
Ian Agol (University of California, Berkeley) . . . drilling and
filling?
Danny Calegari (California Institute of Technology) . . . a
martingale?
Joel Hass (University of California, Davis) . . . an almost normal
surface?
Software demonstrations
Marc Culler (UIC) and Nathan Dunfield (UICU) SnapPy
Research talks
Ian Agol (UC Berkeley) Presentation length and Simon’s
conjecture
Michel Boileau (Universite Paul Sabatier) Graph manifolds which are
integral homology 3-spheres and taut foliations
Marc Culler (University of Illinois at Chicago) Character
varieties, fields, and spectograms of 3-manifolds
Nathan Dunfield (UIUC The Least Spanning Area of a Knot and the
Optimal Bounding Chain Problem
David Gabai (Princeton University) On the topology of ending
laminations space
Cameron Gordon (University of Texas at Austin) L-spaces and
left-orderability
Kazuo Habiro (Kyoto University) Quantum fundamental groups of
3-manifolds
Joel Hass (University of California, Davis) Level n normal
surfaces
Craig Hodgson (The University of Melbourne) Veering triangulations
admit strict angle structures
xiii
William Jaco (Oklahoma State University) Constructing
annular-efficient triangulations
Thang Le (Georgia Institute of Technology) Homology growth, volume,
and Mahler measure
Feng Luo (Rutgers University) Variational principles and rigidity
theorems on triangulated surfaces
Darryl McCullough (University of Oklahoma) Diffeomorphisms and
Heegaard splittings of 3-manifolds
Yoav Moriah (Technion) Heegaard splittings with large subsurface
distances
Walter Neumann (Barnard College, Columbia University) Bilipschitz
geometry of complex surface singularities
Yi Ni (Caltech) Khovanov module and the detection of unlinks
Leonid Polterovich (Chicago/Tel Aviv) Lagrangian knots and
symplectic quasi-measures
Martin Scharlemann (University of California, Santa Barbara) New
examples of manifolds with multiple genus 2 Heegaard
splittings
Abigail Thompson (University of California, Davis) 3-manifolds with
distance two Heegaard splittings
Gang Tian (Beijing University and Princeton University) Symplectic
curvature flow
Genevieve Walsh (Tufts University) Right-angled Coxeter groups,
triangulations of spheres, and hyperbolic orbifolds
Shicheng Wang (Peking University) Graph manifolds have virtually
positive Seifert volume
List of Participants
Mohammed Abouzaid MIT/Clay
Ian Agol UC Berkeley
Christopher Atkinson Temple University
Lashi Bandara ANU
Burzin Bhavnagri The University of Melbourne
John Bland University of Toronto
Michel Boileau Universite Paul Sabatier
Chris Bourne Flinders University
Danny Calegari CalTech
William Cavendish Princeton University
Young Chai Sungkyunkwan University
Sangbum Cho Hanyang University
Julie Clutterbuck ANU
Marc Culler UIC
Ana Janele Dow The University of Melbourne
Nathan Dunfield University of Illinois
Murray Elder The University of Newcastle
Mario Eudave-Munoz UNAM
Cameron Gordon University of Texas at Austin
Kazuo Habiro Kyoto University
Joel Hass UC Davis
Anthony Henderson University of Sydney
Craig Hodgson The University of Melbourne
Neil Hoffman UT Austin
Youngsik Huh Hanyang University
Kazuhiro Ichihara Nihon University
William Jaco Oklahoma State University
Jesse Johnson Oklahoma State University
James Jones The University of Melbourne
Yuichi Kabaya Osaka University
Bryce Kerr Sydney University
Yuya Koda Tohoku University
Andrew Kricker NTU Singapore
Sangyop Lee Chung-Ang University
Feng Luo Rutgers
Hidetoshi Masai Tokyo Institute of Technology
Darryl McCullough University of Oklahoma
Alan McIntosh ANU
Samuel Mellick University of Queensland
PARTICIPANTS xvii
Yi Ni CalTech
Makoto Ozawa Komazawa University
Andrew Percy Monash University
Leonid Polterovich Chicago/Tel Aviv
Arun Ram The University of Melbourne
Matthew Randall ANU
Andrei Ratiu The University of Melbourne
Lawrence Reeves The University of Melbourne
Daniel Rosen University of Chicago
Hyam Rubinstein The University of Melbourne
Martin Scharlemann UC Santa Barbara
Henry Segerman The University of Melbourne
Callum Sleigh The University of Melbourne
John Arthur Snadden UWA
Masakazu Teragaito Hiroshima University
Abigail Thompson UC Davis
Stephan Tillmann The University of Queensland
TriThang Tran The University of Melbourne
Anastasiia Tsvietkova University of Tennesee
Alden Walker CalTech
Shicheng Wang Peking University
xviii PARTICIPANTS
Yoshikazu Yamaguchi Tokyo Institute of Technology
Xi Yao The University of Queensland
George Yiannakopoulos DSTO
PARTICIPANTS xix
Biographical Sketch of Hyam Rubinstein
J. (Joachim) Hyam Rubinstein is a Professor in the Department of
Mathematics and Statistics at the University of Melbourne in
Melbourne, Australia. He was born in 1948 in Melbourne and is the
third of six children, all boys. Hyam and his brothers were
strongly influenced by their mother, who encouraged her sons to
study science and mathematics. All of the brothers were
mathematically minded and keen on chess. Hyam received highest
recognition for academics and mathematics, in particular, before
becoming a teenager, winning the John Braithwaite Scholarship in
1959. He entered Melbourne Boys’ High School and at age 17 years,
topped the State list of matriculation exhibition winners: topping
the general exhibition, with exhibitions in calculus, applied
mathematics, and physics, and winning the B.H.P. Matriculation
Prize. He completed Melbourne Boys’ High School taking the prize
for pure mathematics, physics, and chemistry in his last year of
school.
Hyam then entered Monash University where he majored in pure
mathematics and statistics and earned B.Sc. Honours (First Class)
in 1969. He followed an older brother to University of
California-Berkeley to do graduate work in mathemat- ics. At
Berkeley, Hyam was influenced by the work of John Stallings in
geometric topology and became a student of Stallings. He completed
his thesis and earned his Ph.D. in 1974. While at Berkeley, he was
supported by an IBM Fellowship and received three distinctions in
the qualifying exams. Hyam was by this time married to his wife Sue
and they decided to return to Australia upon the completion of his
doctorate. He accepted a postdoctoral appointment at the University
of Mel- bourne. At the end of his postdoctoral appointment, he
received a contract to stay at Melbourne University and teach, a
position from which he was promoted in the last year to senior
lecturer and he received tenure. In 1982, he was appointed to a
Chair of Mathematics and became a professor at the University of
Melbourne.
During the period prior to Hyam becoming Chair, his predecessor,
Leon Simon, influenced both the Department and Hyam. Through Leon’s
encouragement, Hyam and Jon Pitts started a collaboration that led
to the introduction into 3-manifold topology of sweep outs and
minimax methods from geometric analysis. Hyam’s tremendous breadth
and understanding of mathematics and his generous sharing of ideas
has led to many fruitful collaborations. The early work with Pitts
carried forth in a collaboration on PL minimal surface theory with
William Jaco; later Hyam introduced a polyhedral version of sweep
outs and discovered almost normal surfaces. The latter provided the
methods for Hyam to solve the 3-sphere recogni- tion problem. Hyam
had a long and productive collaboration with Iain Aitchison on
polyhedral differential geometry and another with Marty Scharlemann
on the general structure and methods for comparisons of Heegaard
splittings. He returned to a collaboration with Jaco, both of whom
enjoy triangulations and algorithms
xxi
xxii BIOGRAPHICAL SKETCH OF HYAM RUBINSTEIN
in low-dimensional topology, as well as very good red wine. Hyam
has expanded his interest into a number of collaborations with
young mathematicians, including Ben Burton, Craig Hodgson and
Stephan Tillmann, connecting the geometry and topology of
3-manifolds. In the late 80s, Hyam began a collaboration with
Doreen Thomas on shortest networks, leading to the solution of the
Steiner ratio conjec- ture and the development of a group working
in the design of access to underground mines. This group now
provides consultation around the world on shortest networks in
3-dimensional space and has produced impressive software
introducing their new algorithms to many applications. Hyam also
has an enjoyable collaboration with his son Ben on machine
learning. This collaboration brings geometry and topology into the
science of machine learning, which is Ben’s specialty.
Hyam has earned exceptional recognition. He is a Fellow of the
Australian Academy of Sciences, a Fellow of the American
Mathematical Society, and a Fellow of the Australian Mathematical
Society. He was awarded the Australian Acad- emy’s Hannan Medal for
exceptional mathematical research and the Australian Mathematical
Society’s George Szekeres Medal for outstanding contributions to
the mathematical sciences. He served as president of the Australian
Mathematical Society, Chair of the National Committee for the
Mathematical Sciences, and Chair of the Working Party of the
National Strategic Review of Mathematical Sciences Research in
Australia.
“Geometry and Topology Down Under” is a tribute to Hyam’s
contributions to the algorithmic theory of 3-manifolds, Heegaard
splittings, PL minimal surfaces, sweep outs, almost normal
surfaces, efficient triangulations, and shortest networks. It also
recognizes his influential role throughout a period of exciting and
expansive development in the study and understanding of
low-dimensional topology and 3- manifolds.
Contemporary Mathematics Volume 597, 2013
http://dx.doi.org/10.1090/conm/597/11777
What is an Almost Normal Surface?
Joel Hass
This paper is dedicated to Hyam Rubinstein on the occasion of his
60th birthday.
Abstract. A major breakthrough in the theory of topological
algorithms oc-
curred in 1992 when Hyam Rubinstein introduced the idea of an
almost nor- mal surface. We explain how almost normal surfaces
emerged naturally from the study of geodesics and minimal surfaces.
Patterns of stable and unstable geodesics can be used to
characterize the 2-sphere among surfaces, and similar patterns of
normal and almost normal surfaces led Rubinstein to an algorithm
for recognizing the 3-sphere.
1. Normal Surfaces and Algorithms
There is a long history of interaction between low-dimensional
topology and the theory of algorithms. In 1910 Dehn posed the
problem of finding an algorithm to recognize the unknot [3]. Dehn’s
approach was to check whether the fundamental group of the
complement of the knot, for which a finite presentation can easily
be computed, is infinite cyclic. This led Dehn to pose some of the
first decision problems in group theory, including asking for an
algorithm to decide if a finitely presented group is infinite
cyclic. It was shown about fifty years later that general group
theory decision problems of this type are not decidable [23].
Normal surfaces were introduced by Kneser as a tool to describe and
enumer- ate surfaces in a triangulated 3-manifold [13]. While a
general surface inside a 3-dimensional manifold M can be floppy,
and have fingers and filligrees that wan- der around the manifold,
the structure of a normal surface is locally restricted. When
viewed from within a single tetrahedron, normal surfaces look much
like flat planes. As with flat planes, they cross tetrahedra in
collections of triangles and quadrilaterals. Each tetrahedron has
seven types of elementary disks of this type; four types of
triangles and three types of quadrilaterals. The whole manifold has
7t elementary disk types, where t is the number of 3-simplices in a
triangulation.
Kneser realized that the local rigidity of normal surfaces leads to
finiteness results, and through them to the Prime Decomposition
Theorem for a 3-manifold. This theorem states that a 3-manifold can
be cut open along finitely many 2-spheres into pieces that are
irreducible, after which the manifold cannot be cut further in a
non-trivial way. The idea behind this theorem is intuitively quite
simple: if a
2010 Mathematics Subject Classification. Primary 57N10; Secondary
53A10. Key words and phrases. Almost normal surface, minimal
surface, 3-sphere recognition. Partially supported by NSF grant IIS
1117663.
c©2013 American Mathematical Society
1
2 JOEL HASS
Figure 1. A normal surface intersects a 3-simplex in triangles and
quadrilaterals.
very large number of disjoint surfaces are all uniformly flat, then
some pair of the surfaces must be parallel.
A further advance came in the work of Haken, who gave the first
algorithm for the unknotting problem [6]. Haken realized that a
normal surface could be described by a vector with 7t integer
entries, with each entry describing the number of elementary disks
of a given type. Furthermore the matching of these disks across
faces of a triangulation leads to a collection of integer linear
equations, and this allows application of the techniques of integer
linear programming. In many important cases, the search for a
surface that gives a solution to a topological problem can be
reduced to a search among a finite collection of candidate
surfaces, corresponding to a Hilbert Basis for the space of
solutions to the equations [8]. Problems that can be solved
algorithmically by this approach include:
Problem: UNKNOTTING INSTANCE: A triangulated compact 3-dimensional
manifold M and a collection of edges K in the 1-skeleton of M
QUESTION: Does K bound an embedded disk?
Problem: GENUS INSTANCE: A triangulated compact 3-dimensional
manifold M and a collection of edges K in the 1-skeleton of M and
an integer g QUESTION: Does K bound an embedded orientable surface
of genus g?
Problem: SPLITTING INSTANCE: A triangulated compact 3-dimensional
manifold M and a collection of edges K in the 1-skeleton of M
QUESTION: Does K have distinct components separated by an embedded
sphere?
But one major problem remained elusive.
Problem: 3-SPHERE RECOGNITION INSTANCE: A triangulated
3-dimensional manifold M QUESTION: Is M homeomorphic to the
3-sphere?
Given Perelman’s solution of the 3-dimensional Poincare Conjecture
[16], we know that 3-Sphere Recognition is equivalent to the
following.
Problem: SIMPLY CONNECTED 3-MANIFOLD
WHAT IS AN ALMOST NORMAL SURFACE? 3
INSTANCE: A triangulated compact 3-dimensional manifold M QUESTION:
Is M simply connected?
The 3-Sphere recognition problem has important consequences. Note
for exam- ple that the problem of deciding whether a given
4-dimensional simplicial complex has underlying space which is a
manifold reduces to verifying that the link of each vertex is a
3-sphere, and thus to 3-Sphere Recognition.
In dimension two, the corresponding recognition problem is very
easy. Deter- mining if a surface is homeomorphic to a 2-sphere can
be solved by computing its Euler characteristic. In contrast, for
dimensions five and higher there is no algo- rithm to determine if
a manifold is homeomorphic to a sphere [25], and the status of the
4-sphere recognition problem remains open [15]. The related problem
of fun- damental group triviality is not decidable in manifolds of
dimension four or higher. Until Rubinstein’s work, there was no
successful approach to the triviality problem that took advantage
of the special nature of 3-manifold groups.
For 3-sphere recognition one needs some computable way to
characterize the 3- sphere. Unfortunately all 3-manifolds have zero
Euler characteristic, and no known easily computed invariant that
can distinguish the 3-sphere among manifolds of dimension three.
Approaches developed to characterize spheres in higher dimen- sions
were based on simplifying some description, typically a Morse
function. The simplification process of a Morse function in
dimension three, as given by a Hee- gaard splitting, gets bogged
down in complications. Many attempts at 3-sphere recognition, if
successful, imply combinatorial proofs of the Poincare Conjecture.
Such combinatorial proofs have still not been found. A breakthrough
occurred in the Spring of 1992, at a workshop at the Technion in
Haifa, Israel. Hyam Rubin- stein presented a characterization of
the 3-sphere that was suitable to algorithmic analysis. In a series
of talks at this workshop he introduced a new algorithm that takes
a triangulated 3-manifold and determines whether it is a 3-sphere.
The key new concept was an almost normal surface.
2. What is an almost normal surface?
Almost normal surfaces, as with their normal relatives, intersect
each 3-simplex in M in a collection of triangles or quadrilaterals,
with one exception. In a single 3-simplex the intersection with the
almost normal surface contains, in addition to the usual triangles
or quadrilaterals, either an octagon or a pair of normal disks
connected by a tube, as shown in Figure 2. For Rubinstein’s
3-sphere recognition algorithm, it suffices to consider almost
normal surfaces that contain an octagon disk. Later extensions also
required the second type of local structure, two normal disks
joined by an unknotted tube, one that is parallel to an edge of the
tetrahedron.
Rubinstein argued that an almost normal 2-sphere had to occur in
any trian- gulation of a 3-sphere, and in fact that the search for
the presence or absence of this almost normal 2-sphere could be
used to build an algorithm to recognize the 3-sphere. Shortly
afterwards, Abigail Thompson combined Rubinstein’s ideas with
techniques from the theory of thin position of knots, and gave an
alternate approach to proving that Rubinstein’s algorithm was valid
[24]. The question we address here is the geometrical background
that motivated Rubinstein’s breakthrough.
To describe the ideas from which almost normal surfaces emerged, we
take a diversion into differential geometry and some results in the
theory of geodesics and minimal surfaces. A classical problem asks
which surfaces contain closed, embedded
4 JOEL HASS
Figure 2. Almost normal surfaces intersect one 3-simplex in an
octagon, or two normal disks tubed together.
(or simple) geodesics. The problem is hardest for a 2-sphere, since
for other surfaces a shortest closed curve that is not homotopic to
a point gives an embedded geodesic. A series of results going back
to Poincare establishes that every 2-sphere contains a simple
closed geodesic [2,4,7,12,18]. In fact any 2-sphere always contains
no less than three simple, closed and unstable geodesics. Unstable
means that while each sufficiently short arc of the geodesic
minimizes length among curves connecting its endpoints, the entire
curve can be pushed to either side in a manner that decreases
length. The classic example is an equator of a round sphere, for
which a sub-arc of length shorter than π is length minimizing,
whereas longer arcs can be shortened by a deformation, as can the
whole curve. In Figure 3 we show several differently shaped
2-spheres and indicate unstable geodesics on each of them.
Figure 3. Some unstable geodesics on 2-spheres of various
shapes
A conceptually simple argument shows that unstable geodesics exist
for any Riemannian metric on a 2-sphere, using a minimax argument
that goes back at least to Birkhoff [1]. Starting with a very short
curve, drag it over the 2-sphere until it shrinks to a point on the
other side. Among all such families of curves, look at the family
whose longest curve is as short as possible. This minimax curve
provides an unstable geodesic. It is not hard to show such a curve
exists.
Surfaces other then the 2-sphere do not necessarily contain an
unstable geo- desic. The torus has a flat metric and higher genus
surfaces have hyperbolic metrics, and in these metrics there are no
unstable geodesics. Even the projective plane, the closest
geometric relative of the 2-sphere, has no unstable geodesics in
its ellip- tic metric. Therefore the property of always having an
unstable geodesic, for any metric, characterizes the
2-sphere.
WHAT IS AN ALMOST NORMAL SURFACE? 5
We will need to refine this to develop an algorithm. Any surface
has some met- rics in which there are both stable and unstable
geodesics. So given any fixed Rie- mannian metric on a surface, we
focus on a maximal collection of disjoint separat- ing geodesics,
both stable and unstable. See Figure 4, where unstable geodesics
are drawn as solid curves and stable geodesics as dashed curves. We
assume a “generic” metric on a surface, in which there are only
finitely many disjoint geodesics. Almost all metrics have this
property, which can be achieved by a small perturbation of any
metric [26].
Figure 4. Maximal collections of disjoint separating geodesics on a
2-sphere and a torus. Stable geodesics are shown with broken
curves.
In these examples we see certain patterns among a maximal
collection of disjoint geodesics on a 2-sphere. These are
summarized in the following result.
6 JOEL HASS
Theorem 2.1. Let F be an orientable surface with a generic metric
and G a maximal collection of disjoint, simple, closed and
separating geodesics on F . Then G has the following
properties.
• If F is a 2-sphere then G contains an unstable geodesic. • A
region in F −G whose boundary is a single unstable geodesic is a
disk. • A region in F−G whose boundary is a single stable geodesic
is a punctured
torus. • A region in F − G with two boundary geodesics is an
annulus whose
boundary consists of one stable and one unstable geodesic. • A
region in F−G with three boundary geodesics is a “pair of pants”
whose
boundary consists of three stable geodesics. • No region of F − G
has four or more boundary geodesics.
Proof. The proof applies minimax arguments using the curvature flow
tech- niques developed by Gage, Hamilton, and Grayson [5]. The
curvature flow deforms a curve on a smooth Riemannian surface in
the direction of its curvature vector. Applying this flow to a
family of curves gives a continuous deformation of the entire
family, and decreases the length of each of curve, limiting to a
point or a geodesic [4].
If a region has an unstable geodesic on its boundary, then this
boundary curve can be pushed in slightly and then shrunk by the
curvature flow until it converges to a stable geodesic or to a
point. Thus each region with an unstable geodesic on its boundary
is either a disk or an annulus bounded by one stable and one
unstable geodesic. The boundary curve of a complementary disk
region must be unstable, since shrinking a stable boundary geodesic
to a point gives a family of curves in the disk whose minimax curve
is an unstable geodesic in the interior of the disk. But
complementary regions contain no interior geodesics.
A region bounded by a single stable geodesic cannot contain a
separating essen- tial curve that is not boundary parallel, since
such a curve could be homotoped to a separating geodesic in the
interior of the region. Thus all essential, non-boundary parallel
simple closed curves in the region are non-separating. Such a curve
must exist since the region is not a disk, and so the region must
be a punctured torus.
A minimax argument shows that an annular region bounded by two
stable geodesics has an unstable geodesic separating its two
boundary geodesics. The maximality of G rules out this
configuration.
If a region has two non-homotopic stable geodesics on its boundary,
then we can find a new closed separating curve by tubing the two
boundary geodesics along a shortest arc connecting them within the
region. This new curve can be shortened within the region till it
converges to a third stable geodesic, which must be a third
boundary component. Thus the region is a pair of pants and has
exactly three stable geodesics on its boundary. It follows that no
region has more than three boundary geodesics.
These patterns can be used to distinguish the 2-sphere from other
surfaces. Fix any generic metric on a surface F and let G be a
maximal family of separating, simple, disjoint geodesics.
Theorem 2.2 (Geometric 2-Sphere Characterization). F is a 2-sphere
if G satisfies the following conditions:
WHAT IS AN ALMOST NORMAL SURFACE? 7
• There is at least one unstable geodesic in G. • No complementary
region of F − G has boundary consisting of a single stable
geodesic.
Proof. Suppose that F satisfies these two conditions. Pushing the
unstable geodesic to either side decreases its length. Continuing
to decrease length with the curvature flow, we arrive either at a
stable geodesic or a point. If we arrive at a point then the
unstable geodesic bounds a disk on that side. If we arrive at a
stable geodesic then we consider the region on its other side. If
this region has only one boundary component then the surface is not
a sphere since it contains a punctured torus. If the region has one
other unstable boundary curve then it is an annulus. If the region
has more than two stable boundary curves, then it’s a pair of pants
with three stable boundary geodesics. Continuing across the new
boundary geodesics, we construct a surface from pieces whose dual
graph forms a tree. Unless we encounter a complementary region of F
− G whose boundary has exactly one stable geodesic, the surface F
is a union of annuli, pairs of pants and disks, and these form a
2-sphere.
A very similar characterization carries over to dimension three and
forms the basis of Rubinstein’s 3-sphere recognition algorithm. We
first address the restric- tion of the curves we considered above
to separating curves. One can distinguish separating and
non-separating curves on a surface with homology, and homology can
be efficiently computed from the simplicial structure of a
triangulated manifold. Thus in searching for the 3-sphere we can
immediately rule out any manifold that does not have the same
homology as the 3-sphere. In a homology 3-sphere, every surface
separates. In dimension two, homology itself is enough to
characterize the 2-sphere, though we did not take advantage of this
in our construction. In dimen- sion three, homology computations
alone do not characterize the 3-sphere, but do reduce the
candidates to the class of homology 3-spheres. So we can assume
that we are working in this class and that all surfaces are
separating. In particular we can rule out the possibility that M
contains a non-separating sphere or an embedded projective
plane.
For a characterization of the 3-sphere we look at stable and
unstable mini- mal surfaces instead of geodesics. By 1991
Rubinstein had made two important contributions to the study of
such minimal surfaces in dimension three. Each of these two
contributions played a key role in the creation of the 3-sphere
recognition algorithm.
Rubinstein had worked on the highly non-trivial problem of showing
the ex- istence of minimal representatives for various classes of
surfaces in 3-manifolds. Simon and Smith had shown that the
3-sphere, with any Riemannian metric, con- tains an embedded
minimal 2-sphere [22]. This result was extended by Jost and by
Pitts and Rubinstein [11,17]. In a series of papers Pitts and
Rubinstein devel- oped a program which showed that a very large
class of surfaces in 3-manifolds can be isotoped to be minimal. In
particular, their methods indicated that a strongly irreducible
Heegaard splitting in a 3-manifold always has an unstable minimal
rep- resentative. To show that a 3-sphere, with any Riemannian
metric, contains an unstable minimal 2-sphere, start with a tiny
2-sphere and drag it over the 3-sphere until it shrinks down to a
point on the other side. Among all such families look for the
biggest area 2-sphere in the family and choose a family that makes
this area as small as possible. This minimax construction gives an
unstable minimal
8 JOEL HASS
2-sphere. The existence proof is more subtle than for a geodesic,
but the concepts are similar, and the method extends to give the
following insight. Suppose we take a stable minimal 2-sphere in a
3-sphere and shrink it to a point, after necessar- ily first
enlarging its area. Then among all such families of 2-spheres there
is one whose largest area sphere has smallest area. This minimax
2-sphere is an unstable minimal 2-sphere.
The methods of Pitts-Rubinstein can be used to characterize the
3-ball, simi- larly to the first two conditions of Theorem 2.1. The
theory is considerably harder since there is no simple surface flow
available to decrease area, unlike the curvature flow for curves in
dimension two. Moreover spheres can split into several com- ponents
as their area decreases, unlike curves. However these difficulties
can be overcome [11,17,22].
Suppose B is a 3-manifold:
Geometric 3-Ball Characterization: B is a 3-ball if it satisfies
the following conditions
• The boundary of B is a stable minimal 2-sphere. • The interior of
B contains no stable minimal 2-sphere. • The interior of B contains
an unstable minimal 2-sphere.
The idea of such a 3-Ball Characterization follows the lines of the
two- dimensional case. Suppose that B satisfies the three
assumptions. Then B con- tains an unstable minimal 2-sphere in its
interior. Shrinking this 2-sphere to one side must move it to ∂B,
as otherwise it would get stuck on some stable minimal 2-sphere in
the interior of B. Similarly, shrinking this 2-sphere to the other
side must collapse it to a point, or again it would get stuck on a
stable minimal 2-sphere in the interior of B. Thus B is swept out
by embedded spheres and homeomorphic to a ball.
A similar result characterizes the 3-sphere. Let S be a maximal
family of separating disjoint embedded minimal spheres in M , both
stable and unstable. We are assuming that M is a homology sphere,
so all surfaces separate.
For a generic metric on a 3-manifold M , the collection of disjoint
minimal spheres S is finite. If M contains infinitely many disjoint
minimal spheres, then they can be used to partition M into
infinitely many components. In each compo- nent one can find an
embedded stable minimal sphere by applying the method of
Meeks-Simon-Yau [14]. But stable minimal spheres in M satisfy
uniform bounds on their second fundamental form [21, Theorem 3],
implying a lower bound to the volume between two such spheres
unless they are parallel (meaning that each projects
homeomorphically to the other under the nearest point projection).
An infinite sequence of parallel minimal 2-spheres has a
subsequence converging to a minimal 2-sphere with a Jacobi Field.
But a theorem of White gives the absence of Jacobi fields for a
minimal surface in a generic metric [26].
Geometric 3-Sphere Characterization: M is a 3-sphere if and only if
no complementary region of M − S has boundary consisting entirely
of stable minimal 2-spheres.
Proof. First note that M is homeomorphic to a 3-sphere if and only
if every complementary component X of M − S is a punctured
ball.
WHAT IS AN ALMOST NORMAL SURFACE? 9
Suppose that X is a complementary component of M−S and consider the
case where X has an unstable minimal 2-sphere Σ among its boundary
components. Then we can push Σ in slightly and apply the theorem of
Meeks-Simon and Yau to minimize in its isotopy class [14]. This
gives a collection of stable minimal 2- spheres, that, when joined
by tubes, recover the isotopy class of Σ. We conclude that X is a
punctured ball with exactly one unstable boundary component.
Now suppose that X has all its boundary components stable. We will
show by contradiction that X is not a punctured ball. If it were,
then it could be swept out by a family of 2-spheres. This family
begins with a 2-sphere that tubes together all the boundary
2-spheres of X and ends at a point. By the methods of Simon and
Smith [22], see also [11,17], we obtain an unstable minimal
2-sphere in the interior of X. But this contradicts maximality of
S, so X cannot be a punctured ball.
Together, these cases give the desired characterization.
To translate the geometric characterization into an algorithm, we
need a corre- sponding combinatorial theory that characterizes the
3-sphere among triangulated 3-manifolds. We need to replace the
ideas of Riemannian geometry with PL ver- sions that capture the
relevant ideas. Fortunately, natural PL-approximations to length
and area exist in dimensions two and three. Length is approximated
by the weight, which measures how many times a curve crosses the
edges of a triangula- tion, and area by how many times a surface
intersects edges. Combinatorial length and area can be related to
Riemannian area by taking a series of metrics whose limit has
support on the 1-skeleton.
For curves on a surface, the analog of a geodesic then becomes a
special type of normal curve. A normal curve intersects each
two-simplex in arcs joining distinct edges of the two-simplex, so
that no arc doubles back and has both endpoints on the same edge. A
stable PL-geodesic is defined to be a normal curve for which any
deformation increases weight. For deformations we allow isotopies
of the curve in the surface which are non-transverse to edges or
vertices at finitely many times. An unstable PL-geodesic is a
normal curve that admits a weight decreasing deformation to each of
its two sides. Note that not all normal curves are PL-geodesics. In
the triangulation of the 2-sphere given by a tetrahedron, there are
three unstable PL- geodesics given by quadrilaterals, and
additional unstable PL-geodesics of weight eight and above. A curve
of weight three surrounding a vertex is a normal curve, but not a
PL-geodesic. See Figure 5.
Figure 5. A length four normal curve forms an unstable
PL-geodesic.
10 JOEL HASS
The analogous combinatorial area for surfaces in triangulated
3-manifolds the- ory was investigated in a series of papers by Jaco
and Rubinstein. In their work on PL-minimal surfaces, Jaco and
Rubinstein showed that many of the properties that made minimal
surfaces so useful in studying 3-manifolds still held when us- ing
combinatorial area [9]. For surfaces in 3-manifolds and
deformations of these surfaces that avoid vertices, normal surfaces
play the role of stable minimal sur- faces. The question of which
surfaces take the role of unstable minimal surfaces in the
combinatorial theory was unclear until Rubinstein’s insight that
almost normal surfaces fill this role. Just as unstable geodesics
can be pushed to either side so as to decrease length, and unstable
minimal surfaces can be pushed to either side to decrease area, so
almost normal surfaces can be pushed to either side so as to
decrease weight, or combinatorial area.
These two ingredients, the existence of unstable minimal surfaces
and the con- struction of combinatorial versions of stable and
unstable minimal surfaces, combine to give an algorithm to
recognize the 3-sphere. The characterization of a 3-sphere via its
minimal surfaces can be turned into a characterization via
properties of piecewise linear surfaces, properties that can be
determined by constructing and examining a finite collection of
normal and almost normal surfaces.
3. Recognizing the 3-sphere
Rubinstein’s algorithm is essentially the PL version of the
geometric 3-sphere characterization given above. We take a
candidate manifold M which comes with a fixed triangulation and
first verify that it is a homology 3-sphere. Determining whether M
is homeomorphic to the 3-sphere begins by computing a maximal
family of disjoint, non-parallel normal 2-spheres. There is an
upper bound to the num- ber of simultaneously embedded non-parallel
normal surfaces in M , and a maximal family of normal 2-spheres can
be found with the methods of integer linear pro- gramming. We then
find a maximal family of non-parallel almost normal 2-spheres in
the complement of the family of normal 2-spheres. Let S be the
resulting family of normal and almost normal 2-spheres.
3-Sphere Characterization: M is a 3-sphere if and only if S
satisfies the following conditions:
• There is at least one almost normal sphere in S. • No
complementary region of M − S has boundary consisting of a
single
normal sphere, other than a neighborhood of a vertex.
These conditions can be checked by a finite procedure, and so give
an algorithm. The algorithm for recognizing the 3-sphere proceeds
as follows. One begins
with a collection of 3-simplices and instructions for identifying
their faces in pairs.
• Check that M is a 3-manifold by verifying that the link of each
vertex is a 2-sphere.
• Verify that M has the homology of a 3-sphere. In particular, this
implies that each 2-sphere in M is separating.
• Compute a maximal collection of disjoint non-parallel normal
2-spheres in M . This can be done by solving the normal surface
equations and finding normal 2-spheres among the fundamental
solutions. Then repeat to find a maximal collection of disjoint,
non-parallel, almost-normal 2-spheres in
WHAT IS AN ALMOST NORMAL SURFACE? 11
the complement of the normal 2-spheres. Following Haken, and Jaco-
Tollefson, we can reduce the search for such a family S to a search
within a Hilbert basis of solutions to the integer linear equations
arising from normal surfaces [6,10].
• Cut open the manifold along the maximal collection of disjoint
normal 2-spheres in S and examine each component in turn. An easy
topological argument tells us that M is homeomorphic to a 3-sphere
if and only if every component is homeomorphic to a punctured
3-ball.
• Components with two or more normal boundary 2-spheres are
homeomor- phic to punctured 3-balls. This can be seen by joining
together two normal boundary 2-spheres along a tube that runs
around an edge joining them. Normalizing the resulting 2-sphere
results in either a point or a collection of other boundary
2-spheres. In either case the swept out component is a punctured
ball.
• Components with a single normal 2-sphere on their boundary are
homeo- morphic to a 3-ball if and only if they contain an almost
normal 2-sphere or are neighborhoods (stars) of a vertex. Thompson
showed that the tech- niques of thin position can be used to
establish the existence of almost normal spheres containing one
octagonal disk if the component is a ball [24]. Conversely, if an
almost normal 2-sphere exists then it can be pushed to either side
while reducing its weight, collapsing to a point on one side and a
normal 2-sphere on the other, and establishing that the component
is a ball.
• M is a 3-sphere if and only if every component with a single
normal 2- sphere on its boundary contain an almost normal 2-sphere
or is a vertex neighborhood.
The structure of the algorithm is very similar to the 2-sphere
characterization described above. The characterization of the
various complementary regions is also similar to that in dimension
two. The evolution of a curve by curvature is replaced by a
normalization procedure in which a surface deforms to become normal
or al- most normal.
Remark. There are differences between the characterizations used in
the smooth and PL settings. In the smooth setting, an unstable
minimal 2-sphere always ex- ists in the interior of a punctured
ball whose boundary consists of stable minimal 2-spheres. In
contrast, a region in a triangulated 3-manifold bounded by two or
more normal 2-spheres and containing no normal 2-spheres in its
interior is always a punctured ball.
4. Conclusion
Rubenstein’s work on the existence of minimal surfaces in
3-manifolds and on PL-minimal surface theory naturally led him to
the concept of an almost normal surface. Almost normal surfaces are
now widely recognized as powerful tools to apply in multiple areas
of 3-manifold theory.
Table 1 summarizes some correspondences between the worlds of
Riemannian manifolds with their minimal submanifolds and of
triangulated manifolds with their normal and almost normal
submanifolds.
12 JOEL HASS
Smooth Riemannian Manifolds Combinatorial Triangulated
Manifolds
Geodesic Normal curve
Unstable minimal surface Almost normal surface
Flow by mean curvature Normalization
A smooth S3 contains an unstable minimal S2 A PL S3 contains an
almost normal S2
∂X a stable S2 and int(X) contains ∂X a normal S2 and int(X)
contains an unstable S2, no stable S2 an almost normal S2, no
normal S2
=⇒ X = B3 =⇒ X = B3
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Department of Mathematics, University of California, Davis,
California 95616
E-mail address:
[email protected]
The Ergodic Theory of Hyperbolic Groups
Danny Calegari
Abstract. These notes are a self-contained introduction to the use
of dy- namical and probabilistic methods in the study of hyperbolic
groups. Most of this material is standard; however some of the
proofs given are new, and some results are proved in greater
generality than have appeared in the literature.
Contents
1. Introduction 2. Hyperbolic groups 3. Combings 4. Random walks
Acknowledgments References
1. Introduction
These are notes from a minicourse given at a workshop in Melbourne
July 11– 15 2011. There is little pretension to originality; the
main novelty is firstly that we give a new (and much shorter) proof
of Coornaert’s theorem on Patterson–Sullivan measures for
hyperbolic groups (Theorem 2.5.4), and secondly that we explain how
to combine the results of Calegari–Fujiwara in [8] with that of
Pollicott–Sharp [35] to prove central limit theorems for quite
general classes of functions on hyperbolic groups (Corollary 3.7.5
and Theorem 3.7.6), crucially without the hypothesis that the
Markov graph encoding an automatic structure is ergodic.
A final section on random walks is much more cursory.
2. Hyperbolic groups
2.1. Coarse geometry. The fundamental idea in geometric group
theory is to study groups as automorphisms of geometric spaces, and
as a special case, to study the group itself (with its canonical
self-action) as a geometric space. This is accomplished most
directly by means of the Cayley graph construction.
2010 Mathematics Subject Classification. Primary 20F10, 20F32,
20F67, 37D20, 60B15, 60J50, 68Q70.
c©2013 American Mathematical Society
15
16 DANNY CALEGARI
Definition 2.1.1 (Cayley graph). Let G be a group and S a (usually
finite) generating set. Associated to G and S we can form the
Cayley graph CS(G). This is a graph with vertex set G, and with an
edge from g to gs for all g ∈ G and s ∈ S.
The action of G on itself by (left) multiplication induces a
properly discontin- uous action of G on CS(G) by simplicial
automorphisms.
If G has no 2-torsion, the action is free and properly
discontinuous, and the quotient is a wedge of |S| circles XS . In
this case, if G has a presentation G = S | R we can think of CS(G)
as the covering space of XS corresponding to the subgroup of the
free group FS normally generated by R, and the action of G on CS(G)
is the deck group of the covering.
Figure 1. The Cayley graph of F2 = a, b | with generating set S =
{a, b}
We assume the reader is familiar with the notion of a metric space,
i.e. a space X together with a symmetric non-negative real-valued
function dX on X×X which vanishes precisely on the diagonal, and
which satisfies the triangle inequality dX(x, y) + dX(y, z) ≥ dX(x,
z) for each triple x, y, z ∈ X. A metric space is a path metric
space if for each x, y ∈ X, the distance dX(x, y) is equal to the
infimum of the set of numbers L for which there is a 1-Lipschitz
map γ : [0, L] → X sending 0 to x and L to y. It is a geodesic
metric space if it is a path metric space and if the infimum is
achieved on some γ for each pair x, y; such a γ is called a
geodesic. Finally, a metric space is proper if closed metric balls
of bounded radius are compact (equivalently, for each point x the
function d(x, ·) : X → R is proper).
The graph CS(G) can be canonically equipped with the structure of a
geodesic metric space. This is accomplished by making each edge
isometric to the Euclidean unit interval. If S is finite, CS(G) is
proper. Note that G itself inherits a subspace metric from CS(G),
called the word metric. We denote the word metric by dS , and
define |g|S (or just |g| if S is understood) to be dS(id, g).
Observe that dS(g, h) = |g−1h|S = |h−1g|S and that |g|S is the
length of the shortest word in elements of S and their inverses
representing the element g.
The most serious shortcoming of this construction is its dependence
on the choice of a generating set S. Different choices of
generating set S give rise to
THE ERGODIC THEORY OF HYPERBOLIC GROUPS 17
different spaces CS(G) which are typically not even homeomorphic.
The standard way to resolve this issue is to coarsen the geometric
category in which one works.
Definition 2.1.2. Let X, dX and Y, dY be metric spaces. A map f : X
→ Y (not assumed to be continuous) is a quasi-isometric map if
there are constants K ≥ 1, ε ≥ 0 so that
K−1dX(x1, x2)− ε ≤ dY (f(x1), f(x2)) ≤ KdX(x1, x2) + ε
for all x1, x2 ∈ X. It is said to be a quasi-isometry if further
f(X) is a net in Y ; that is, if there is some R so that Y is equal
to the R-neighborhood of f(X).
One also uses the terminology K, ε quasi-isometric map or K, ε
quasi-isometry if the constants are specified. Note that a K, 0
quasi-isometric map is the same thing as a K bilipschitz map. The
best constant K is called the multiplicative constant, and the best
ε the additive constant of the map.
We denote the R-neighborhood of a set Σ by NR(Σ). Hence a
quasi-isometry is a quasi-isometric map for which Y = NR(f(X)) for
some R.
Remark 2.1.3. It is much more common to use the terminology
quasi-isometric embedding instead of quasi-isometric map as above;
we consider this terminology misleading, and therefore avoid
it.
Lemma 2.1.4. Quasi-isometry is an equivalence relation.
Proof. Reflexivity and transitivity are obvious, so we must show
symmetry. For each y ∈ Y choose x ∈ X with dY (y, f(x)) ≤ R (such
an x exists by definition) and define g(y) = x. Observe dY (y,
fg(y)) ≤ R by definition. Then
dX(g(y1), g(y2)) ≤ KdY (fg(y1), fg(y2)) + Kε ≤ KdY (y1, y2) + K(ε +
2R)
Similarly,
dX(g(y1), g(y2)) ≥ K−1dY (fg(y1), fg(y2))−K−1ε ≥ K−1dY (y1,
y2)−K−1(ε+2R)
proving symmetry.
Note that the compositions fg and gf as above move points a bounded
distance. One can define a category in which objects are
equivalence classes of metric spaces under the equivalence relation
generated by thickening (i.e. isometric inclusion as a net in a
bigger space), and morphisms are equivalence classes of
quasi-isometric maps, where two maps are equivalent if their values
on each point are a uniformly bounded distance apart. In this
category, quasi-isometries are isomorphisms. In particular, the set
of quasi-isometries of a metric space X, modulo maps that move
points a bounded distance, is a group, denoted QI(X), which only
depends on the quasi-isometry type of X. Determining QI(X), even
for very simple spaces, is typically extraordinarily
difficult.
Example 2.1.5. A metric space X, dX is quasi-isometric to a point
if and only if it has bounded diameter. A Cayley graph CS(G) (for S
finite) is quasi-isometric to a point if and only if G is
finite.
Example 2.1.6. If S and T are two finite generating sets for a
group G then the identity map from G to itself is a quasi-isometry
(in fact, a bilipschitz map) of G, dS to G, dT . For, there are
constants C1 and C2 so that dT (s) ≤ C1 for all s ∈ S, and dS(t) ≤
C2 for all t ∈ T , and therefore C−1
2 dT (g, h) ≤ dS(g, h) ≤ C1dT (g, h).
18 DANNY CALEGARI
Because of this, the quasi-isometry class of G, dS is independent
of the choice of finite generating set, and we can speak
unambiguously of the quasi-isometry class of G.
The Schwarz Lemma connects the geometry of groups to the geometry
of spaces they act on.
Lemma 2.1.7 (Schwarz Lemma). Let G act properly discontinuously and
co- compactly by isometries on a proper geodesic metric space X.
Then G is finitely generated by some set S, and the orbit map G → X
sending g to gx (for any x ∈ X) is a quasi-isometry from G, dS to
X.
Proof. Since X is proper and G acts cocompactly there is an R so
that GNR(x) = X. Note that Gx is a net, since every point of X is
contained in some translate gB and is therefore within distance R
of gx.
Let B = N2R+1(x). Since G acts properly discontinuously, there are
only finitely many g in G for which gB∩B is nonempty; let S be the
nontrivial elements of this set.
Now, if g, h ∈ G are arbitrary, let γ be a geodesic in X from gx to
hx. Pa- rameterize γ by arclength, and for each integer i ∈ (0,
|γ|) let gi be such that dX(gix, γ(i)) ≤ R. Then g−1
i gi+1 ∈ S and therefore
dS(g, h) = |g−1h| ≤ |γ|+ 1 = d(gx, hx) + 1
which shows incidentally that S generates G. Conversely, if L :=
dS(g, h) and gi is a sequence of elements with g0 = g and
gL = h and each g−1 i gi+1 ∈ S, then there is a path γi from gix to
gi+1x of length
at most 4R + 2, and the concatenation of these paths certifies
that
d(gx, hx) ≤ (4R + 2)|g−1h| = (4R + 2)dS(g, h)
This completes the proof of the lemma. Example 2.1.8. If G is a
group and H is a subgroup of finite index, then G
and H are quasi-isometric (for, both act properly discontinuously
and cocompactly on CS(G)). Two groups are said to be commensurable
if they have isomorphic subgroups of finite index; the same
argument shows that commensurable groups are quasi-isometric.
Example 2.1.9. Any two regular trees of (finite) valence≥ 3 are
quasi-isometric; for, any such tree admits a cocompact action by a
free group of finite rank, and any two free groups of finite rank
are commensurable.
Example 2.1.10. The set of ends of a geodesic metric space is a
quasi-isometry invariant. A famous theorem of Stallings [39] says
that a finitely generated group with more than one end splits over
a finite subgroup; it follows that the property of splitting over a
finite subgroup is a quasi-isometry invariant.
Finiteness of the edge groups (in a splitting) is detected
quasi-isometrically by the existence of separating compact subsets.
Quasi-isometry can further detect the finiteness of the vertex
groups, and in particular one observes that a group is
quasi-isometric to a free group if and only if it is virtually
free.
Example 2.1.11. Any two groups that act cocompactly and properly
discontin- uously on the same space X are quasi-isometric. For
example, if M1, M2 are closed Riemannian manifolds with isometric
universal covers, then π1(M1) and π1(M2) are
THE ERGODIC THEORY OF HYPERBOLIC GROUPS 19
quasi-isometric. It is easy to produce examples for which the
groups in question are not commensurable; for instance, a pair of
closed hyperbolic 3-manifolds M1, M2 with different invariant trace
fields (see [27]).
Remark 2.1.12. In the geometric group theory literature, Lemma
2.1.7 is of- ten called the “Milnor–Svarc (or Svarc-Milnor) Lemma”;
“Svarc” here is in fact the well-known mathematical physicist
Albert Schwarz; it is our view that the orthog- raphy “Svarc” tends
to obscure this. Actually, the content of this Lemma was first
observed by Schwarz in the early 50’s and only rediscovered 15
years later by Milnor at a time when the work of Soviet
mathematicians was not widely disseminated in the west.
2.2. Hyperbolic spaces. In a geodesic metric space a geodesic
triangle is just a union of three geodesics joining three points in
pairs. If the three points are x, y, z we typically denote the
(oriented) geodesics by xy, yz and zx respectively; this notation
obscures the possibility that the geodesics in question are not
uniquely determined by their endpoints.
Definition 2.2.1. A geodesic metric space X, dX is δ-hyperbolic if
for any geodesic triangle, each side of the triangle is contained
in the δ-neighborhood of the union of the other two sides. A metric
space is hyperbolic if it is δ-hyperbolic for some δ.
One sometimes says that geodesic triangles are δ-thin.
Figure 2. A δ-thin triangle; the gray tubes have thickness δ.
Example 2.2.2. A tree is 0-hyperbolic.
Example 2.2.3. Hyperbolic space (of any dimension) is δ-hyperbolic
for a uniform δ.
Example 2.2.4. If X is a simply-connected complete Riemannian
manifold with curvature bounded above by some K < 0 then X is
δ-hyperbolic for some δ depending on K.
20 DANNY CALEGARI
Definition 2.2.5. A geodesic metric space X is CAT(K) for some K if
tri- angles are thinner than comparison triangles in a space of
constant curvature K. This means that if xyz is a geodesic triangle
in X, and x′y′z′ is a geodesic triangle in a complete simply
connected Riemannian manifold Y of constant curvature K with edges
of the same lengths, and φ : xyz → x′y′z′ is an isometry on each
edge, then for any w ∈ yz we have dX(x, w) ≤ dY (x′, φ(w)).
The initials CAT stand for Cartan–Alexandrov–Toponogov, who made
sub- stantial contributions to the theory of comparison
geometry.
Example 2.2.6. From the definition, a CAT(K) space is δ-hyperbolic
whenever the complete simply connected Riemannian 2-manifold of
constant curvature K is δ-hyperbolic. Hence a CAT(K) space is
hyperbolic if K < 0.
Example 2.2.7. Nearest point projection to a convex subset of a
CAT(K) space with K ≤ 0 is distance nonincreasing. Therefore the
subspace metric and the path metric on a convex subset of a CAT(K)
space agree, and such a subspace is itself CAT(K).
Thinness of triangles implies thinness of arbitrary polygons.
Example 2.2.8. Let X be δ-hyperbolic and let abcd be a geodesic
quadrilateral. Then either there are points on ab and cd at
distance ≤ 2δ or there are points on ad and bc at distance ≤ 2δ, or
possibly both.
Figure 3. Two ways that a quadrilateral can be thin
The number of essentially distinct ways in which an n-gon can be
thin is equal to the nth Catalan number. By cutting up a polygon
into triangles and examining the implications of δ-thinness for
each triangle, one can reason about the geometry of complicated
configurations in δ-hyperbolic space.
Lemma 2.2.9. Let X be δ-hyperbolic, let γ be a geodesic
segment/ray/line in X, and let p ∈ X. Then there is a point q on γ
realizing the infimum of distance from p to points on γ, and
moreover for any two such points q, q′ we have dX(q, q′) ≤
4δ.
Proof. The existence of some point realizing the infimum follows
from the properness of d(p, ·) : γ → R, valid for any geodesic in
any metric space.
THE ERGODIC THEORY OF HYPERBOLIC GROUPS 21
Let q, q′ be two such points, and if d(q, q′) > 4δ let q′′ be
the midpoint of the segment qq′, so d(q, q′′) = d(q′′, q′) > 2δ.
Without loss of generality there is r on pq with d(r, q′′) ≤ δ
hence d(r, q) > δ. But then
d(p, q′′) ≤ d(p, r) + d(r, q′′) ≤ d(p, r) + δ < d(p, r) + d(r,
q) = d(p, q)
contrary to the fact that q minimizes the distance from p to points
on γ. Lemma 2.2.9 says that there is an approximate “nearest point
projection”
map π from X to any geodesic γ (compare with Example 2.2.7). This
map is not continuous, but nearby points must map to nearby points,
in the sense that d(π(x), π(y)) ≤ d(x, y) + 8δ.
We would now like to show that the property of being hyperbolic is
pre- served under quasi-isometry. The problem is that the property
of δ-hyperbolicity is expressed in terms of geodesics, and
quasi-isometries do not take geodesics to geodesics.
A quasigeodesic segment/ray/line is the image of a segment/ray/line
in R under a quasi-isometric map. For infinite or semi-infinite
intervals this definition has content; for finite intervals this
definition has no content without specifying the constants
involved. Hence we can talk about a K, ε quasigeodesic
segment/ray/line.
Lemma 2.2.10 (Morse lemma). Let X, dX be a proper δ-hyperbolic
space. Then for any K, ε there is a constant C (depending in an
explicit way on K, ε, δ) so that any K, ε quasigeodesic γ is within
Hausdorff distance C of a genuine geodesic γg. If γ has one or two
endpoints, γg can be chosen to have the same endpoints.
Proof. If γ is noncompact, it can be approximated on compact
subsets by finite segments γi. If we prove the lemma for finite
segments, then a subsequence of the γg
i , converging on compact subsets, will limit to γg with the
desired properties (here is where we use properness of X). So it
suffices to prove the lemma for γ a segment.
In this case choose any γg with the same endpoints as γ. We need to
estimate the Hausdorff distance from γ to γg. Fix some constant C
and suppose there are points p, p′ on γ that are both distance C
from γg, but d(r, γg) ≥ C for all r on γ between p and p′. Choose
pi a sequence of points on γ and qi a sequence of points on γg
closest to the pi so that d(qi, qi+1) = 11δ.
Consider the quadrilateral pipi+1qi+1qi. By Example 2.2.8 either
there are close points on pipi+1 and qiqi+1, or close points on
piqi and pi+1qi+1 (or possibly both). Suppose there are points ri
on piqi and ri+1 on pi+1qi+1 with d(ri, ri+1) ≤ 2δ. Then any
nearest point projections of ri and ri+1 to γg must be at most
distance 10δ apart. But qi and qi+1 are such nearest point
projections, by definition, and satisfy d(qi, qi+1) = 11δ. So it
must be instead that there are points ri on pipi+1 and si on qiqi+1
which are at most 2δ apart. But this means that d(pi, pi+1) ≥ 2C −
4δ, so the length of γ between p and p′ is at least (2C − 4)d(q,
q′)/11δ where q, q′ are points on γ closest to p, p′. On the other
hand, d(p, p′) ≤ 2C + d(q, q′). Since γ is a K, ε quasigeodesic, if
d(q, q′) is big enough, we get a uniform bound on C in terms of K,
ε, δ. The remaining case where d(q, q′) is itself uniformly bounded
but C is unbounded quickly leads to a contradiction.
Corollary 2.2.11. Let Y be δ-hyperbolic and let f : X → Y be a K, ε
quasi- isometry. Then X is δ′-hyperbolic for some δ. Hence the
property of being hyperbolic is a quasi-isometry invariant.
22 DANNY CALEGARI
Proof. Let Γ be a geodesic triangle in X with vertices a, b, c.
Then the edges of f(Γ) are K, ε quasigeodesics in Y , and are
therefore within Hausdorff distance C of geodesics with the same
endpoints. It follows that every point on f(ab) is within distance
2C + δ of f(ac) ∪ f(bc) and therefore every point on ab is within
distance K(2C + δ) + ε of ac ∪ bc.
The Morse Lemma lets us promote quasigeodesics to (nearby)
geodesics. The next lemma says that quasigeodesity is a local
condition.
Definition 2.2.12. A path γ in X is a k-local geodesic if the
subsegments of length ≤ k are geodesics. Similarly, γ is a k-local
K, ε quasigeodesic if the subsegments of length ≤ k are K, ε
quasigeodesics.
Lemma 2.2.13 (k-local geodesics). Let X be a δ-hyperbolic geodesic
space, and let k > 8δ. Then any k-local geodesic is K, ε
quasigeodesic for K, ε depending explicitly on δ.
More generally, for any K, ε there is a k and constants K ′, ε′ so
that any k-local K, ε quasigeodesic is a K ′, ε′
quasigeodesic.
Proof. Let γ be a k-local geodesic segment from p to q, and let γg
be any geodesic from p to q. Let r be a point on γ furthest from
γg, and let r be the midpoint of an arc r′r′′ of γ of length 8δ. By
hypothesis, r′r′′ is actually a geodesic. Let s′ and s′′ be points
on γg closest to r′ and r′′. The point r is within distance 2δ
either of γg or of one of the sides r′s′ or r′′s′′. If the latter,
we would get a path from r to s′ or s′′ shorter than the distance
from r′ or r′′, contrary to the definition of r. Hence the distance
from r to γg is at most 2δ, and therefore γ is contained in the 2δ
neighborhood of γg.
Now let π : γ → γg take points on γ to closest points on γg. Since
π moves points at most 2δ, it is approximately continuous. Since γ
is a k-local geodesic, the map π is approximately monotone; i.e. if
pi are points on γ with d(pi, pi+1) = k moving monotonely from one
end of γ to the other, then d(π(pi), π(pi+1)) ≥ k− 4δ and the
projections also move monotonely along γ. In particular, d(pi, pj)
≥ (k − 4δ)|i− j| and π is a quasi-isometry. The constants involved
evidently depend only on δ and k, and the multiplicative constant
evidently goes to 1 as k gets large.
The more general fact is proved similarly, by using Lemma 2.2.10 to
promote local quasigeodesics to local geodesics, and then back to
global quasigeodesics.
2.3. Hyperbolic groups. Corollary 2.2.11 justifies the following
definition:
Definition 2.3.1. A group G is hyperbolic if CS(G) is δ-hyperbolic
for some δ for some (and hence for any) finite generating set
S.
Example 2.3.2. Free groups are hyperbolic, since their Cayley
graphs (with respect to a free generating set) are trees which are
0-hyperbolic.
Example 2.3.3. Virtually free groups, being precisely the groups
quasi-isometric to trees, are hyperbolic. A group quasi-isometric
to a point or to R is finite or virtu- ally Z respectively; such
groups are called elementary hyperbolic groups; all others are
nonelementary.
Example 2.3.4. Fundamental groups of closed surfaces with negative
Euler characteristic are hyperbolic. By the uniformization theorem,
each such surface can be given a hyperbolic metric, exhibiting π1
as a cocompact group of isometries of the hyperbolic plane.
THE ERGODIC THEORY OF HYPERBOLIC GROUPS 23
Example 2.3.5. A Kleinian group is a finitely generated discrete
subgroup of the group of isometries of hyperbolic 3-space. A
Kleinian group G is is convex cocompact if it acts cocompactly on
the convex hull of its limit set (in the sphere at infinity). Such
a convex hull is CAT(−1), so a convex cocompact Kleinian group is
hyperbolic. See e.g. [28] for an introduction to Kleinian
groups.
Lemma 2.3.6 (invariant quasiaxis). Let G be hyperbolic. Then there
are finitely many conjugacy classes of torsion elements (and
therefore a bound on the order of the torsion) and there are
constants K, ε so that for any nontorsion element g there is a K, ε
quasigeodesic γ invariant under g on which g acts as
translation.
Proof. Let g ∈ G be given. Consider the action of g on the Cayley
graph CS(G). The action is simplicial, so p → d(p, gp) has no
strict local minima in the interior of edges, and takes integer
values at the vertices (which correspond to elements of G). It
follows that there is some h for which d(h, gh) is minimal, and we
can take h to be an element of G (i.e. a vertex). If d(h, gh) = k
> 8δ then we can join h to gh by a geodesic σ and let γ =
∪ig
iσ. Note that g acts on γ by translation through distance k; since
this is the minimum distance that g moves points of G, it follows
that γ is a k-local geodesic (and therefore a K, ε quasigeodesic by
Lemma 2.2.13). Note in this case that g has infinite order.
Otherwise there is h moved a least distance by g so that d(h, gh) ≤
8δ. Since G acts cocompactly on itself, there are only finitely
many conjugacy classes of elements that move some point any
uniformly bounded distance, so if g is torsion we are done. If g is
not torsion, its orbits are proper, so for any T there is an N so
that d(h, gNh) > T ; choose T (and N) much bigger than some
fixed (but big) n. Let γ be a geodesic from h to gNh. Then for any
0 ≤ i ≤ n the geodesic giγ has endpoints within distance 8δn of the
endpoints of γ. On the other hand, |γ| = T 8δn so γ contains a
segment σ of length at least T − 16δn − O(δ) such that giσ is
contained in the 2δ neighborhood of γ for 0 ≤ i ≤ n. To see this,
consider the quadrilateral with successive vertices h, gNh, gi+Nh
and gih. Two nonadjacent sides must contain points which are at
most 2δ apart. Since N i, the sides must be γ and giγ. We find σ
and giσ in the region where these two geodesics are close.
Consequently, for any p ∈ σ the sequence p, gp, · · · , gnp is a K,
ε quasigeodesic for some uniform K, ε independent of n. In
particular there is a constant C (in- dependent of n) so that d(p,
gip) ≥ iC for 0 ≤ i ≤ n, and therefore the infinite sequence gip
for i ∈ Z is an (nC)-local K, ε quasigeodesic. Since K, ε is fixed,
if n is big enough, this infinite sequence is an honest K ′, ε′
quasigeodesic invariant under g, by Lemma 2.2.13. Here K ′, ε′
depends only on δ and G, and not on g.
Lemma 2.3.6 can be weakened considerably, and it is frequently
important to study actions which are not necessarily cocompact on
δ-hyperbolic spaces which are not necessarily proper. The
quasigeodesic γ invariant under g is called a quasiaxis. Quasiaxes
in δ-hyperbolic spaces are (approximately) unique:
Lemma 2.3.7. Let G be hyperbolic, and let g have infinite order.
Let γ and γ′ be g-invariant K, ε quasigeodesics (i.e. quasiaxes for
g). Then γ and γ′ are a finite Hausdorff distance apart, and this
finite distance depends only on K, ε and δ. Consequently the
centralizer C(g) is virtually Z.
Proof. Let p ∈ γ and p′ ∈ γ′ a closest point to p. Since g acts on
both γ and γ′ cocompactly, there is a constant C so that every
point in γ or γ′ is within C
24 DANNY CALEGARI
from some point in the orbit of p or p′. This implies that the
Hausdorff distance from γ to γ′ is at most 2C + d(p, p′); in
particular, this distance is finite.
Pick two points on γ very far away from each other; each is
distance at most 2C + d(p, p′) from γ′, and therefore most of the
geodesic between them is within distance 2δ of the geodesic between
corresponding points on γ′. But γ and γ′
are themselves K, ε quasigeodesic, and therefore uniformly close to
these geodesics. Hence some points on γ are within a uniformly
bounded distance of γ′, and therefore all points on γ are.
If h commutes with g, then h must permute the quasiaxes of g.
Therefore h takes points on any quasiaxis γ for g to within a
bounded distance of γ. Hence C(g), thought of as a subset of G, is
quasiisometric to a quasiaxis (that is to say, to R), and is
therefore virtually Z.
This shows that a hyperbolic group cannot contain a copy of Z ⊕ Z
(or, for that matter, the fundamental group of a Klein bottle).
This is more subtle than it might seem; Z ⊕ Z can act freely and
properly discontinuously by isometries on a proper δ-hyperbolic
space — for example, as a parabolic subgroup of the isometries of
H3.
Example 2.3.8. If M is a closed 3-manifold, then π1(M) is
hyperbolic if and only if it does not contain any Z⊕ Z subgroup.
Note that this includes the possi- bility that π1(M) is elementary
hyperbolic (for instance, finite). This follows from Perelman’s
Geometrization Theorem [31,32].
If g is an isometry of any metric space X, the translation length
of g is the limit τ (g) := limn→∞ dX(p, gnp)/n for some p ∈ X. The
triangle inequality implies that the limit exists and is
independent of the choice of p. Moreover, from the definition, τ
(gn) = |n|τ (g) and τ (g) is a conjugacy invariant.
Lemma 2.3.6 implies that for G acting on itself, τ (g) = 0 if and
only if g has finite (and therefore bounded) order. Consequently a
hyperbolic group cannot contain a copy of a Baumslag–Solitar group;
i.e. a group of the form BS(p, q) := a, b | bapb−1 = aq. For, we
have already shown hyperbolic groups do not contain Z ⊕ Z, and this
rules out the case |p| = |q|, and if |p| = |q| then for any
isometric action of BS(p, q) on a metric space, τ (a) = 0.
By properness of CS(G) and the Morse Lemma, there is a constant N
so that for any g ∈ G the power gN has an invariant geodesic axis
on which it acts by translation. It follows that τ (g) ∈ Q, and in
fact ∈ 1
NZ; this cute observation is due to Gromov [20].
2.4. The Gromov boundary. Two geodesic rays γ, γ′ in a metric space
X are asymptotic if they are a finite Hausdorff distance apart. The
property of being asymptotic is an equivalence relation, and the
set of equivalence classes is the Gromov boundary, and denoted ∂∞X.
If X is proper and δ-hyperbolic, and x is any basepoint, then every
equivalence class contains a ray starting at x. For, if γ is a
geodesic ray, and gi