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Bounded cohomology and \(\ell ^ 1-\hom o\log y\) of surfaces. (English) Zbl 0568.55002
Generalizing a result of R. Brooks and C. Series [ibid. 23, 29-36 (1984; Zbl 0523.55011)], the author shows that the dimension of the second bounded cohomology group of a surface with non-amenable fundamental group is the cardinality of continuum. He shows this at least in two different ways. These simplify the proof of Brooks-Series. The one is a geometrical method using the perturbation of the hyperbolic metric of the surface and the other is a group homological method using the Euler class. In the course of the proof, he introduces the \(\ell^ 1\)- homology group of surfaces and obtains the same result on the dimension of the second \(\ell^ 1\)-homology group.
Reviewer: T.Mizutani

55N35 Other homology theories in algebraic topology
57T30 Bar and cobar constructions
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