- This book is a comprehensive introduction to the theory of
stable commutator length,
an important subfield of quantitative topology, with substantial connections to
2-manifolds, dynamics, geometric group theory, bounded cohomology,
symplectic topology, and many other subjects.
We use constructive methods whenever possible, and focus on fundamental
and explicit examples. We give a self-contained presentation of several
foundational results in the theory, including Bavard's Duality Theorem, the
Spectral Gap Theorem, the Rationality Theorem, and the Central Limit Theorem.
The contents should be accessible to any mathematician
interested in these subjects, and
are presented with a minimal number of prerequisites,
but with a view to applications
in many areas of mathematics.
- The book has been published by the Mathematical Society of Japan as volume 20 in their MSJ Memoirs series.
The .pdf text of the book will continue to be
freely available, but I strongly encourage you to buy a copy. The books are very reasonably priced, and may be ordered from World Scientific (price US $31) or Amazon.
- Download (corrected) final version of monograph
- scallop, a command line scl calculator, and wallop, with a graphical interface, can be downloaded (as well as some other related programs: scabble, sss, etc) from Alden Walker's github page.
- A cumulative list of corrections to the printed version of monograph; note that the electronic version is kept up to date to include corrections as I become aware of them (24th November 2009)
- Thanks to Aaron Abrams, Marc Burger, Jean-Louis Clerc, Matthew Day, Benson Farb, David Fisher, Steven Frankel,
Koji Fujiwara, Ilya Kapovich,
Dieter Kotschick, Justin Malestein, Fedor Manin, Curt McMullen, Geoff Mess, Assaf Naor,
Andy Putman, Pierre Py, Peter Sarnak, Alden Walker, Anna Wienhard,
Dave Witte-Morris and Dongping Zhuang
for additions and corrections. Special thanks to Jason Manning for extensive
comments (especially on Chapters 1 and 2), and for many useful conversations
about numerous technical points. Special thanks to Sadayoshi Kojima for soliciting the book for MSJ in the first place, and to Takashi Tsuboi for acting as handling editor.
Finally, special thanks to Shigenori Matsumoto
for his meticulous refereeing, which led to
vast improvements throughout the book.
- Zentralblatt MATH review by Athanase Papadopoulos
by Athanase Papadopoulos
- Table of contents:
- Chapter 1: Surfaces
- Triangulating surfaces
- Hyperbolic surfaces
- Chapter 2: Stable commutator length
- Commutator length and stable commutator length
- Bounded cohomology
- Bavard's Duality Theorem
- Stable commutator length as a norm
- Extremal quasimorphisms
- Further properties
- Chapter 3: Hyperbolicity and spectral gaps
- Hyperbolic manifolds
- Spectral Gap Theorem
- Hyperbolic groups
- Counting quasimorphisms
- Mapping class groups
- Chapter 4: Free and surface groups
- The Rationality Theorem
- Geodesics on surfaces
- Small cancellation theory
- Chapter 5: Irrationality and dynamics
- Stein-Thompson groups
- Groups with few quasimorphisms
- Braid groups and transformation groups
- Chapter 6: Combable functions and ergodic theory
- An example
- Groups and automata
- Combable functions
- Counting quasimorphisms
- Patterson-Sullivan measures