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On planar Cayley graphs and Kleinian groups.
- Source :
-
Transactions of the American Mathematical Society . Jul2020, Vol. 373 Issue 7, p4649-4684. 36p. - Publication Year :
- 2020
-
Abstract
- Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X ⊆ S2. We prove that G admits such an action that is in addition co-compact, provided we can replace X by another surface Y ⊆ S2. We also prove that if a group H has a finitely generated Cayley (multi-) graph C equivariantly embeddable in S2, 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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *PLANAR graphs
*CAYLEY graphs
*GENERATING functions
*HOMEOMORPHISMS
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 373
- Issue :
- 7
- Database :
- Academic Search Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 143784671
- Full Text :
- https://doi.org/10.1090/tran/8026