# Hyperbolic group with boundary $S^1$ implies virtually Fuchsian via bounded cohomology?

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.

• I'm not sure exactly what sort of proof you had in mind, but a possible natural idea is to imitate a rigidity theorem of Jungreis which works in higher dimensions. mathscinet.ams.org/mathscinet-getitem?mr=1452185 He proves that there is a unique measure on ideal simplices which realizes the Gromov norm. However, this won't work in 2 dimensions. For each faithful discrete representation of a fuchsian group, one gets a measure cycle on ideal triangles which is Haar measure of $PSL(2,\mathbb{R})$, and realizes the Gromov norm. So I'm not sure if one can recover a $PSL(2,\mathbb{R})$ rep. – Ian Agol Nov 28 '18 at 4:49
• As far as "Groups acting on circles" is concerned, here is a link perso.ens-lyon.fr/ghys/articles/groupscircle.pdf – Denis Chaperon de Lauzières Nov 28 '18 at 5:54
• @DenisChaperondeLauzières Thanks, it looks like Ghys specifically mentions that he doesn't talk about it in the paper. – Paul Plummer Nov 28 '18 at 5:59
• The (frequent) way of stating the result with "virtually" is a bit unfortunate... indeed every hyperbolic group $G$ has a maximal normal finite subgroup $W(G)$ and the result is that if $G/W(G)$ is topologically a circle then $G/W(G)$ is Fuchsian (possibly orientation-reversing), i.e. a cocompact lattice in $PGL_2(R)$. The "virtual" statement is also true, in the sense that there exists a finite index subgroup intersecting $W(G)$ trivially in this case, since surface groups are "$2$-good" in the sense of Serre. – YCor Nov 28 '18 at 6:05
• @IanAgol The rough idea I was thinking was to understand "convergence implies conjugate to Fuchsian"(which I think is the hard part) by trying to understand bounded Euler classes of convergence groups. If there is a way to recognize if a bounded Euler class is equal to on $PSL(2,\mathbb R)$ rep then, by some theorems of Ghys, these should be conjugate representations. – Paul Plummer Nov 28 '18 at 6:11