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**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 PoincarĂ© 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.

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 PoincarĂ© 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.

Reviewer: Athanase Papadopoulos (Strasbourg)

### MSC:

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 |