boundary of word-hyperbolic group

Let $G$ be a word-hyperbolic group and let $\partial G$ be its (Gromov) boundary. Do there exist criteria that imply that all non-trivial finite order elements of $G$ act fixed-point freely on $\partial G$?

• Surely not, because the boundary is a tree and so every finite order element fixes a point? (I believe it forms a tree as this is where JSJ-decompositions come from, if I have interpreted Bowditch's paper on cut points and canonical splitting's correctly). – user1729 Aug 9 '13 at 14:45
• The boundary is certainly not always a tree. If the group acts properly and cocompactly on $\mathbb{H}^{n}$, the boundary is homeomorphic to $S^{n-1}$. – user68316 Aug 9 '13 at 14:48
• Hmm. Can what you say happen in the one-ended case? – user1729 Aug 9 '13 at 14:49
• If $\partial G$ is connected, then $G$ has one end. – user68316 Aug 9 '13 at 15:01
• Oh yeah, Bowditch mentions that! – user1729 Aug 9 '13 at 15:37