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- W4288351317 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." @default.
- W4288351317 created "2022-07-29" @default.
- W4288351317 creator A5061366186 @default.
- W4288351317 date "2019-05-16" @default.
- W4288351317 modified "2023-09-29" @default.
- W4288351317 title "On planar Cayley graphs and Kleinian groups" @default.
- W4288351317 doi "https://doi.org/10.48550/arxiv.1905.06669" @default.
- W4288351317 hasPublicationYear "2019" @default.
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