Suppose $X$ is a proper, geodesic, $ \delta $-hyperbolic, metric space. Let $\partial X $ denote the Gromov boundary of $X$.
It is known that isometries of $X$ are exactly one of three kinds:
- Elliptic (or orbit of $ x \in X $ is bounded.
- Parabolic (has exactly one fixed point in the boundary)
- Loxodromic (has exactly two fixed points in the boundary)
My Questions:
Can an elliptic isometry fix a point in the boundary?
Can the stabilizer of a parabolic fixed point contain an elliptic element?
Can the stabilizer of a loxodromic fixed point have an elliptic element?