Let G be a word-hyperbolic group acting on its boundary, which is homeomorphic to $S^n$ (n-sphere), effectively. Does this imply that G acts on the boundary as a convergence group of $S^n$? 


If this is true in general, is it easy to see, at least for n = 1 or 2? 



convergence action on the boundary of hyperbolic groups
Bowditch proved much more. Namely, if a group $\Gamma$ acts properly discontinuously on a $\delta$-hyperbolic space $X$, then $\Gamma$ acts as a convergence group on $\partial X$. See Lemma 1.11 of his paper 


B.H. Bowditch,
Convergence groups and configuration spaces,
in ``Geometric Group Theory Down Under, proceedings of a Special Year in Geometric Group Theory, Canberra, Australia'' (ed. J.Cossey, C.F.Miller III, W.D.Neumann, M.Shapiro), de Gruyter (1999), 23-54.


which is available [here](http://www.warwick.ac.uk/~masgak/papers/bhb-convergence.dvi.gz). To get the result you want, consider the action of $\Gamma$ on its Cayley graph.


