Bounded cohomology and l1-homology of surfaces

Surfaces et cohomologie born�e

“…En fait, R. Brooks et C. Series [3] et S. Mitsumatsu [9] ont montr~ par des m~thodes combinatoires que H~(nl (X), I~)~-H~(X, R) est un espace de dimension infinie non d~nombrable. En fait, R. Brooks et C. Series [3] et S. Mitsumatsu [9] ont montr~ par des m~thodes combinatoires que H~(nl (X), I~)~-H~(X, R) est un espace de dimension infinie non d~nombrable.…”

The cup product of Brooks quasimorphisms

“…In all other cases, f w is a quasimorphism and defines a non-trivial class β w ∈ H 2 b (F, R) unless w is conjugated to a power of a letter. The space spanned by all these β w is infinite-dimensional [1] [8] and is dense in H 2 b (F, R) for a suitable topology of pointwise convergence [4, 5.7]. (Following Brooks, we allow overlaps when counting occurrences, whilst other authors do not; see [5, p. 251] for the density in our setting.…”

An invitation to bounded cohomology

“…Johnson proves already in [89, 2.8] that the free group F 2 admits a nontrivial quasimorphism. This was considerably generalised and it is now known that EH 2 b (G, R) is infinite-dimensional for (non-elementary) free groups, surface groups, Gromov-hyperbolic groups, free products (R. Brooks [11], Brooks-Series [12], Y. Mitsumatsu [111], Barge-Ghys [5], Epstein-Fujiwara [46], K. Fujiwara [49], [50], R. Grigorchuk [66]); generalising all the previous cases, for all groups acting on a Gromov-hyperbolic metric space in a weakly proper way (Bestvina-Fujiwara [8]; see also U. Hamenstädt [74]). Moreover, J. Manning [102, 4.29] [9], GambaudoGhys [58] and P. Py [131].…”