On planar Cayley graphs and Kleinian groups

Abstract

Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X ⊆ 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 ⊆ S 2 .We also prove that if a group H has a finitely generated Cayley (multi-)graph C covariantly 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.In the orientation preserving case, the groups of Theorem 1.1 coincide, as abstract groups, with the Kleinian function groups mentioned above: Corollary 1.2. A finitely generated group Γ admits a faithful, properly discontinuous (co-compact) action by orientation-preserving homeomorphisms on a planar surface if and only if it is isomorphic to a Kleinian function group.Corollary 1.2 can be deduced from [26, THEOREM 4], which essentially says that the groups of (B) coincide in the orientation-preserving case with the Kleinian function groups. We will give an alternative proof bypassing Ahlfors' finiteness theorem.Our proof of Corollary 1.2 makes use of a classical theorem of Maskit, saying that if p :S → S is a regular covering of a topologically finite surface S,

2‐complexes with unique embeddings in 3‐space

“…We remark that infinite versions of Theorem 1.1 have been obtained by Imrich [13] and Thomassen and Richter [16]. The latter was used by the first author [9] in order to understand discrete group actions on non‐compact surfaces. Theorem 3.4 may find similar applications on discrete group actions on $$\mathbb {R}^3$$.…”

“…To prove the backward direction of this statement, one can use Theorem 1.1 to extend the canonical action of $$\Gamma$$ on a Cayley graph embedded in $$\mathbb {S}^2$$ to an action on $$\mathbb {S}^2$$. See [9] for more. This paper was motivated by the quest (currently in progress) to extend such results to groups acting on $$\mathbb {S}^3$$ or other 3‐manifolds.…”

$2$-complexes with unique embeddings in 3-space

“…We remark that infinite versions of Theorem 1.1 have been obtained by Imrich [11] and Thomassen & Richter [14]. The latter was used by the first author [8] in order to understand discrete group actions on non-compact surfaces. Theorem 3.4 may find similar applications on discrete group actions on R 3 .…”

“…To prove the backward direction of this statement, one can use Theorem 1.1 to extend the canonical action of Γ on a Cayley graph embedded in S 2 to an action on S 2 . See [8] for more. This paper was motivated by the quest (currently in progress) to extend such results to groups acting on S 3 or other 3-manifolds.…”