# Group theory Does a hyperbolic group with the limit \$ S ^ 1 \$ imply virtually Fuchsian through limited cohomology?

Question: Is there an approach to $$partial G cong S ^ 1$$ involves virtually Fuchsian using limited cohomology of $$mathrm {Homeo ^ +} (S ^ 1)$$? If not, is there a reason to believe that it would not work, or perhaps much more difficult than the known tests?

I recently decided that I would like to try to learn a proof of the theorem that for a hyperbolic group $$G$$ with $$partial G cong S ^ 1$$ It implies that the group is virtually Fuchsian. From what I understand, the test is to show that the convergence group acts on $$S ^ 1$$ conjugate with Fuchsian groups, and then show that hyperbolic groups act at their limits as convergence groups.

Something I would like to learn more about is the applications of limited cohomology. I understand that it is useful to determine things like the conjugation of representations in $$mathrm {Homeo} ^ + (S ^ 1)$$. Therefore, I suppose (perhaps naively) that it would be useful or an alternative approach to the previous question. If it existed it would be a great application …

I'm having a hard time finding information about this (and I can not find a copy of Groups that act on the circle. by Ghys for some reason, which is where I would think it would be discussed if it were a plausible approach), so I guess there are not many connections, but maybe someone has thought of this before or is an expert who can see why it is limited. Cohomology would not be useful.