  Bowditch characterized relative hyperbolicity in terms of group actions on
fine hyperbolic graphs with finitely many edge orbits and finite edge
stabilizers. In this paper, we define generalized fine actions on hyperbolic
graphs, in which the peripheral subgroups are allowed to stabilize finite
sub-graphs rather than stabilizing a point. Generalized fine actions are useful
for studying groups that act relatively geometrically on a CAT(0) cube complex,
which were recently defined by the first two authors. Specifically, we show
that a group acting relatively geometrically on a CAT(0) cube complex admits a
generalized fine action on the one-skeleton of the cube complex. For
generalized fine actions, we prove a criterion for relative quasiconvexity as
subgroups that cocompactly stabilize quasi-convex sub-graphs, generalizing a
result of Martinez-Pedroza and Wise in the setting of fine hyperbolic graphs.
As an application, we obtain a characterization of boundary separation in
generalized fine graphs and use it to prove that Bowditch boundary points in
relatively geometric actions are always separated by a hyperplane stabilizer.
