scl. (English) Zbl 1187.20035

MSJ Memoirs 20. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-53-2/pbk). xii, 209 p. (2009).
The three letters in the title are a shorthand for “stable commutator length”. We recall the definition.
Let \(G\) be a group. A ‘commutator’ in \(G\) is an element in \(G\) of the form \(ghg^{-1}h^{-1}\), for some \(g\) and \(h\) in \(G\). The ‘commutator subgroup’ \([G,G]\) of \(G\) is the subgroup generated by the commutators.
Let \(a\) be an element of the commutator subgroup \([G,G]\). The ‘commutator length’ of \(a\), denoted by \(\mathrm{cl}(a)\), is the least number of commutators in \(G\) whose product is equal to \(a\).
If \(a\) is not in \([G,G]\), then one sets, by convention, \(\mathrm{cl}(a)=\infty\).
Let \(a\) be again element of the commutator subgroup \([G,G]\). The ‘stable commutator length’ of \(a\), denoted by \(\mathrm{scl}(a)\), is defined as \[ \mathrm{scl}(a)=\lim_{n\to\infty}\frac{\mathrm{cl}(a^n)}{n}. \] From the fact that the function \(n\mapsto\mathrm{cl}(a^n)\) is nonnegative and subadditive, the above limit always exists.
The history of stable commutator length can be traced back to the work of Poincaré on rotation numbers.
The subject of the monograph under review is stable commutator length and its applications in geometry. The fields of applications that are considered include combinatorial group theory, group representation, \(3\)-manifold topology, surface homeomorphisms, hyperbolic groups, the study of outer space, and applications in probability theory and dynamical system theory. Examples of topics considered include Bavard Duality Theorem that relates stable commutator length and quasimorphisms, the relation with Thurston’s norm on second homology, the work of Burger and Monod on bounded cohomology with applications to rigidity, and the results of Calegari and Fujiwara on stable commutator length in hyperbolic groups.
The author brings together successfully various aspects of all these fields in a coherent way, and the result is a self-contained and an extremely interesting book which will be useful to specialists in geometry, combinatorial group theory or dynamics, and to graduate students in any one of these fields.


20F12 Commutator calculus
20-02 Research exposition (monographs, survey articles) pertaining to group theory
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
20F69 Asymptotic properties of groups
20F05 Generators, relations, and presentations of groups
20J05 Homological methods in group theory
57M07 Topological methods in group theory
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
37E45 Rotation numbers and vectors
37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
60C05 Combinatorial probability
Full Text: DOI Link