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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On planar Cayley graphs and Kleinian groups

Author: Agelos Georgakopoulos
Journal: Trans. Amer. Math. Soc. 373 (2020), 4649-4684
MSC (2010): Primary 05C10, 57M60, 57M07, 57M15
Published electronically: March 16, 2020
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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$ equivariantly 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.

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Additional Information

Agelos Georgakopoulos
Affiliation: Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom

Keywords: Planar Cayley graphs, equivariant embedding, Kleinian groups, properly discontinuous actions, planar surface, Freudenthal compactification
Received by editor(s): May 2, 2019
Received by editor(s) in revised form: September 2, 2019
Published electronically: March 16, 2020
Additional Notes: The author was supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 639046)
Article copyright: © Copyright 2020 Agelos Georgakopoulos