geometry and topology down under

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
Editors
597
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 articles, 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 researchers 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 introduced 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 decomposition 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.
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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 collaborations 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 geometric 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 number 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 diffeomorphisms. 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 mathematics. 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 recognition 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
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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 conjecture 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 normal 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 example 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 algorithm 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 fundamental 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 dimensions 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 triangulation 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 geodesic. 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.
We will need to refine this to develop an algorithm. Any surface has some metrics 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 separating 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 techniques 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 essential 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:
• 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 dimension 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 minimal 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 existence of minimal representatives for various classes of surfaces in 3-manifolds. Simon and Smith had shown that the 3-sphere, with any Riemannian metric, contains 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 developed 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 necessarily 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, similarly 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 components 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 contains 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 component 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.
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 corresponding combinatorial theory that characterizes the 3-sphere among triangulated 3-manifolds. We need to replace the ideas of Riemannian geometry with PL versions 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 triangulation, 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 theory 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 using combinatorial area [9]. For surfaces in 3-manifolds and deformations of these surfaces that avoid vertices, normal surfaces play the role of stable minimal surfaces. 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 construction 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 number 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 programming. 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
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 homeomorphic 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 homeomorphic 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 techniques 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 almost 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 exists 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
References
[1] G. Birkhoff, Dynamical Systems, AMS, 1927. [2] Christopher B. Croke, Poincare’s problem and the length of the shortest closed geodesic on a
convex hypersurface, J. Differential Geom. 17 (1982), no. 4, 595–634. MR683167 (84f:58034)
[3] M. Dehn, Uber die Topologie des dreidimensionalen Raumes, Math. Ann. 69 (1910), no. 1, 137–168, DOI 10.1007/BF01455155 (German). MR1511580
[4] Matthew A. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), no. 2, 285–314. MR906392 (89b:53005)
[5] Matthew A. Grayson, Shortening embedded curves, Ann. of Math. (2) 129 (1989), no. 1, 71–111, DOI 10.2307/1971486. MR979601 (90a:53050)
[6] Wolfgang Haken, Theorie der Normalflachen, Acta Math. 105 (1961), 245–375 (German). MR0141106 (25 #4519a)
[7] Joel Hass and Peter Scott, Shortening curves on surfaces, Topology 33 (1994), no. 1, 25–43, DOI 10.1016/0040-9383(94)90033-7. MR1259513 (94k:57025)
[8] Joel Hass, Jeffrey C. Lagarias, and Nicholas Pippenger, The computational complexity of knot and link problems, J. ACM 46 (1999), no. 2, 185–211, DOI 10.1145/301970.301971. MR1693203 (2000g:68056)
[9] William Jaco and J. Hyam Rubinstein, PL minimal surfaces in 3-manifolds, J. Differential Geom. 27 (1988), no. 3, 493–524. MR940116 (89e:57009)
[10] William Jaco and Jeffrey L. Tollefson, Algorithms for the complete decomposition of a closed 3-manifold, Illinois J. Math. 39 (1995), no. 3, 358–406. MR1339832 (97a:57014)
[11] Jurgen Jost, Embedded minimal surfaces in manifolds diffeomorphic to the three-dimensional ball or sphere, J. Differential Geom. 30 (1989), no. 2, 555–577. MR1010172 (90j:58031)
[12] Wilhelm Klingenberg, Closed geodesics on Riemannian manifolds, CBMS Regional Confer- ence Series in Mathematics, vol. 53, Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1983. MR714330 (85g:58029)
[13] H. Kneser, “Geschlossene Flachen in dreidimensionalen Mannigfaltigkeiten”, Jahresbericht
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spheres, and manifolds with positive Ricci curvature, Ann. of Math. (2) 116 (1982), no. 3, 621–659, DOI 10.2307/2007026. MR678484 (84f:53053)
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[16] G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three- manifolds, arXiv:math.DG/0307245 [math.DG]. (2003).
[17] Jon T. Pitts and J. H. Rubinstein, Existence of minimal surfaces of bounded topological type in three-manifolds, (Canberra, 1985), Proc. Centre Math. Anal. Austral. Nat. Univ., vol. 10, Austral. Nat. Univ., Canberra, 1986, pp. 163–176. MR857665 (87j:49074)
[18] H. Poincare, Sur les lignes godsiques sur les surfaces convexes, Trans. Amer. Math. Soc. 17, 237-274 (1909).
[19] Joachim H. Rubinstein, An algorithm to recognize the 3-sphere, 2 (Zurich, 1994), Birkhauser, Basel, 1995, pp. 601–611. MR1403961 (97e:57011)
[20] J. H. Rubinstein, Polyhedral minimal surfaces, Heegaard splittings and decision problems for 3-dimensional manifolds, Geometric topology (Athens, GA, 1993), AMS/IP Stud. Adv. Math., vol. 2, Amer. Math. Soc., Providence, RI, 1997, pp. 1–20. MR1470718 (98f:57030)
[21] Richard Schoen, Estimates for stable minimal surfaces in three-dimensional manifolds, Semi- nar on minimal submanifolds, Ann. of Math. Stud., vol. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 111–126. MR795231 (86j:53094)
[22] F. Smith, On the existence of embedded minimal 2-spheres in the 3-sphere, endowed with
an arbitrary Riemannian metric, Ph.D. thesis, supervisor L. Simon, University of Melbourne (1982).
[23] John Stillwell, The word problem and the isomorphism problem for groups, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 33–56, DOI 10.1090/S0273-0979-1982-14963-1. MR634433 (82m:20039)
[24] Abigail Thompson, Thin position and the recognition problem for S3, Math. Res. Lett. 1 (1994), no. 5, 613–630. MR1295555 (95k:57015)
[25] I. A. Volodin, V. E. Kuznecov, and A. T. Fomenko, The problem of the algorithmic discrim- ination of the standard three-dimensional sphere, Uspehi Mat. Nauk 29 (1974), no. 5(179), 71–168 (Russian). Appendix by S. P. Novikov. MR0405426 (53 #9219)
[26] Brian White, The space of minimal submanifolds for varying Riemannian metrics, Indiana Univ. Math. J. 40 (1991), no. 1, 161–200, DOI 10.1512/iumj.1991.40.40008. MR1101226 (92i:58028)
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 dynamical 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 discontinuous 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 cocompactly 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 discontinuously 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
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 triangles 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.
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 virtually 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.
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 (independent 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 possibility 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

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