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**Bounded cohomology and non-uniform perfection of mapping class groups.**
*(English)*
Zbl 0987.57004

It is well known that the mapping class group of a closed orientable surface of genus \(g\geq 3\) is perfect. The authors prove that it is not uniformly perfect. That is, there is no positive integer \(N\) such that every element of the mapping class group can be written as a product of at most \(N\) commutators. This follows from their main result: the number of commutators needed to express the \(k\)th power of a Dehn twist about a separating simple closed curve grows linearly with \(k\). In order to prove this, an upper bound to the number of separating vanishing cycles in a Lefschetz fibration is given in terms of the number of nonseparating vanishing cycles and the genera of the base and of the fiber. From previously known results it is then concluded that the natural map from the second bounded cohomology to the ordinary cohomology is not injective for the mapping class group and for the Torelli group. Similar results were also obtained independently by the reviewer in [M. Korkmaz, arXiv:math.GT/0012162, (2000)]. It is now known that the second bounded cohomology of the mapping class group is infinite dimensional [M. Bestvina and K. Fujiwara, Geom. Topol. 6, 69-89 (2002)].

Reviewer: Mustafa Korkmaz (Ankara)