**Question:** Is there an approach to $\partial G \cong S^1$ implies virtually Fuchsian using bounded cohomology of $\mathrm{Homeo^+} (S^1)$? If not is there a reason to believe it wouldn't work, or maybe just a lot harder than known proofs?

I recently decided I would like to attempt to learn a proof of the theorem that for a hyperbolic group $G$ with $\partial G \cong S^1$ implies that the group is virtually Fuchsian. From what I understand the proof goes through showing that convergence group acting on $S^1$ conjugate to Fuchsian groups, and then show that hyperbolic group act on their boundary as convergence groups.

Something else I would like to learn more about is applications of bounded cohomology. It is my understanding that it is useful in determining things like conjugacy of representations into $\mathrm{Homeo}^+(S^1)$. So, I would guess (perhaps naively) that it would be useful or an alternative approach to the above question. If there was it would be a cool application...

I am having a hard time finding information on this (and can't find a copy of *Groups acting on the circle* by Ghys for some reason which is where I would think it would be discussed if it was a plausible approach) so I am guessing there isn't much connections, but maybe someone has thought about this before or is an expert who can see why bounded cohomology would not be useful.

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