ON PLANAR CAYLEY GRAPHS AND KLEINIAN GROUPS

Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X subset of S-2. We prove that G admits such an action that is in addition co-compact, provided we can replace X by another surface Y subset of S-2. We also prove that if a group H has a finitely generated Cayley (multi-) graph C equivariantly embeddable in S-2, then 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.