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Let $\Gamma$ be a non elementary hyperbolic group acting on the Gromov boundary $\partial\Gamma$. Let $a \in \Gamma$ be a torsion element i.e $<a>$ is finite. Does $a$ fix every element of $\partial\Gamma$?
I know that that every element $a$ is either a parabolic element in which case it has one...
In mathematics, a convergence group or a discrete convergence group is a group
Γ
{\displaystyle \Gamma }
acting by homeomorphisms on a compact metrizable space
M
{\displaystyle M}
in a way that generalizes the properties of the action of Kleinian group by Möbius transformations on the ideal boundary
S
2
{\displaystyle \mathbb {S} ^{2}}
of the hyperbolic 3-space
...
