Abstract. In this paper, we give a proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe the norms which we constructed are equivalent to the biinvariant word norm studied by Brandenbursky and Kȩdra.
We consider a generalized Gambaudo-Ghys construction on bounded cohomology and prove its injectivity. As a corollary, we prove that the third bounded cohomology of the group of area-preserving diffeomorphisms on the 2-disc is infinite-dimensional. We also prove similar results for the case of the 2-sphere and the 2-torus.
Let (S, ω) be a closed orientable surface whose genus l is at least two. Then we provide an obstruction for commuting symplectomorphisms in terms of the flux homomorphism. More precisely, we show that for every nonnegative integer n and for every homomorphism α :For the proof, we show the following two keys: a refined version of the non-extendability of Py's Calabi quasimorphism µ P on Ham c (S, ω), and an extension theorem of Ĝ-invariant quasimorphisms on G for a group Ĝ and a normal subgroup G with certain conditions.We also pose the conjecture that the cup product of the fluxes of commuting symplectomorphisms is trivial.
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