On planar Cayley graphs and Kleinian groups. (English) Zbl 1442.05086
Summary: 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.
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.
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 57M60 | Group actions on manifolds and cell complexes in low dimensions |
| 57M07 | Topological methods in group theory |
| 57M15 | Relations of low-dimensional topology with graph theory |
Keywords:
planar Cayley graphs; equivariant embedding; Kleinian groups; properly discontinuous actions; planar surface; Freudenthal compactificationReferences:
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