For every countable weakly hyperbolic group, classifying general type actions on hyperbolic spaces is either smooth (isotropic case) or E_Kσ complete (anisotropic case), and all complexity levels are realized.
Gromov, Hyperbolic groups, Essays in Group Theory, MSRI Series 8, (S.M
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Classifying group actions on hyperbolic spaces
For every countable weakly hyperbolic group, classifying general type actions on hyperbolic spaces is either smooth (isotropic case) or E_Kσ complete (anisotropic case), and all complexity levels are realized.