Let
G
G
be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface
X
⊆
S
2
X \subseteq \mathbb {S}^2
. We prove that
G
G
admits such an action that is in addition co-compact, provided we can replace
X
X
by another surface
Y
⊆
S
2
Y \subseteq \mathbb {S}^2
.
We also prove that if a group
H
H
has a finitely generated Cayley (multi-) graph
C
C
equivariantly embeddable in
S
2
\mathbb {S}^2
, then
C
C
can be chosen so as to have no infinite path on the boundary of a face.
The proofs of these facts are intertwined, and the classes of groups they define coincide. In the orientation-preserving case they are exactly the (isomorphism types of) finitely generated Kleinian function groups. We construct a finitely generated planar Cayley graph whose group is not in this class.
In passing, we observe that the Freudenthal compactification of every planar surface is homeomorphic to the sphere.