# On planar Cayley graphs and Kleinian groups

## TL;DR

This paper characterizes finitely generated groups acting on planar surfaces and their Cayley graphs, linking geometric group actions with Kleinian groups, and constructs examples outside this class.

## Contribution

It establishes conditions under which groups admit co-compact planar actions and relates these to Kleinian groups, also providing a counterexample of a planar Cayley graph outside this class.

## Key findings

- Groups with planar Cayley graphs correspond to Kleinian function groups.
- Every planar surface's Freudenthal compactification is homeomorphic to the sphere.
- Constructed a planar Cayley graph with a group not in the Kleinian class.

## Abstract

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