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I think that the question is self explanatory. The only example that comes to my mind is the characterization of hyperbolic groups given by Mineyev ("Straightening and bounded cohomology"). There is also a characterization of amenability of groups using bounded cohomology. However, amenability doesn't seems to me a geometric property.

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    $\begingroup$ Among the many equivalent characterizations of amenability is the existence of a sequence of Folner sets, which is a geometric characterization. $\endgroup$
    – Lee Mosher
    Apr 22, 2015 at 12:44

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There is a long strand of literature relating hyperbolicity of groups and of group actions to second bounded cohomology of groups.

The first result in this theory is Brooks' proof that free groups of positive rank have second bounded cohomology of uncountably infinite dimension. After further progress eventually Epstein and Fujiwara proved the same conclusion for all word hyperbolic groups.

Then Fujiwara wrote a series of papers studying non-properly discontinuous group actions on hyperbolic spaces, giving conditions on such actions under which the second bounded cohomology is of uncountable dimension.

After further progress, Bestvina and Fujiwara proved that every subgroup of a surface mapping class group is either virtually abelian or has second bounded cohomology of uncountable dimension. One of their main tools was the theorem of Masur and Minsky on Gromov hyperbolicity of the curve complex of a surface, and they used and deeloped related results about the geometry and dynamics of actions of subgroups of mapping class groups on curve complexes.

A further interesting feature of this story is that the machinery developed by Bestvina and Fujiwara, for purposes of computing second bounded cohomology of subgroups of mapping class groups, played back into the general study of group actions on hyperbolic spaces, leading to the theory of "acylindrical group actions" developed first by Bowditch and more recently by Osin, and the very recent theory of "hierarchically hyperbolic groups" developed by Behrstock, Hagen, and Sisto.

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  • $\begingroup$ Your examples are quite interesting. All of them relate bounded cohomology with (some subclasses of) hyperbolic groups or generalizations. Are there examples genuinely unrelated to hyperbolic groups and generalizations (as the example of amenable groups)? $\endgroup$
    – fritz
    Apr 23, 2015 at 9:54

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