On planar Cayley graphs and Kleinian groups

2019/05/16 by Georgakopoulos, Agelos · 1 citation
#05C10 #57M07 #57M15 #57M60 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT)

paper · doi:10.48550/arxiv.1905.06669

Abstract

Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X ⊆ \mathbbS². We prove that G admits such an action that is in addition co-compact, provided we can replace X by another surface Y ⊆ \mathbbS². We also prove that if a group H has a finitely generated Cayley (multi-)graph C covariantly embeddable in \mathbbS², 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.

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