Consider the surface group $\Gamma=\langle a,b,c,d\mid [a,b][c,d]=1\rangle$: it is a Gromov hyperbolic group; its Gromov boundary $\partial\Gamma$ is homeomorphic to $S^1$ (the unit circle). I would like to define a family of subsets of $\partial\Gamma$ as follows: fix $x,y\in\Gamma$. Then $$ U(x,y):=\left\{\xi\in\partial\Gamma\mid\text{ there exists a geodesic }g\text{ in }\Gamma\text{ starting from }x,\text{ passing through }y\text{ and s.t. }g(\infty)=\xi\right\}. $$

- Does this definition make sense? Is $U(x,y)$ trivial?
- Is $U(x,y)$ open?
- Is $U(x,y)$ connected?

shadowof an element $y$ of a hyperbolic group is commonly used. (One may as well take $x=1$.) See, for instance, arXiv:1111.0029v2 . $\endgroup$