The Planar Cayley Graphs are Effectively Enumerable I: Consistently Planar Graphs

Georgakopoulos, Agelos; Hamann, Matthias

doi:10.1007/s00493-019-3763-3

Cited by 6 publications

(11 citation statements)

References 27 publications

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“…Thus function groups form a subfamily (proper, as shown in this paper) of the planar groups, i.e. the groups having planar Cayley graphs, which are studied in recent work by the author and others [1,10,11,13,17,18,19,24,34,42,47]. We will prove that each finitely generated function group admits a planar Cayley graph having no infinite facial path, by possibly allowing loops and parallel edges.…”

Section: Introductionmentioning

confidence: 84%

“…For the proof of Theorem 1.3, we use (C) and the main result of [18]. The latter states that every covariantly planar Cayley graph G is the 1-skeleton of a Cayley complex Z, which complex can be mapped into S 2 in such a way that (a) the restriction of the map to G is covariant, and (b) the images of any two 2-cells of Z are nested , i.e.…”

Section: Proof Ideasmentioning

confidence: 99%

“…We will use the notion of planar presentations, and the associated almost planar Cayley complexes, from [18,19], about which we need to prove a couple of additional facts. We start by recalling some terminology.…”

Section: Relationship To Planar Groups and Planar Discontinuous Groupsmentioning

confidence: 99%

“…The groups of Theorem 1.1 can be described by a certain kind of group presentation, which allows them to be effectively enumerated [18]. Thus we also obtain an effective enumeration of the isomorphism types of Kleinian function groups.…”

Section: Introductionmentioning

confidence: 99%

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On planar Cayley graphs and Kleinian groups

Georgakopoulos

2020

Trans. Amer. Math. Soc.

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Let G be a finitely generated group acting faithfully and properly discontinuously by homeomorphisms on a planar surface X ⊆ 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 ⊆ S 2 .We also prove that if a group H has a finitely generated Cayley (multi-)graph C covariantly embeddable in 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.In the orientation preserving case, the groups of Theorem 1.1 coincide, as abstract groups, with the Kleinian function groups mentioned above: Corollary 1.2. A finitely generated group Γ admits a faithful, properly discontinuous (co-compact) action by orientation-preserving homeomorphisms on a planar surface if and only if it is isomorphic to a Kleinian function group.Corollary 1.2 can be deduced from [26, THEOREM 4], which essentially says that the groups of (B) coincide in the orientation-preserving case with the Kleinian function groups. We will give an alternative proof bypassing Ahlfors' finiteness theorem.Our proof of Corollary 1.2 makes use of a classical theorem of Maskit, saying that if p :S → S is a regular covering of a topologically finite surface S,

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Section: Introductionmentioning

confidence: 84%

Section: Proof Ideasmentioning

confidence: 99%

Section: Relationship To Planar Groups and Planar Discontinuous Groupsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On planar Cayley graphs and Kleinian groups

Georgakopoulos

2020

Trans. Amer. Math. Soc.

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“…The presentations of Table 1 have a special structure that is related to the embedding of the corresponding Cayley graph, and yield geometric information about the corresponding Cayley complex. The ideas of this paper are used in [16] to prove that every planar Cayley graph admits such a presentation. This solves the aforementioned problem of [8,9] asking for an effective enumeration; see Section 1.4.…”

Section: Introduction 1overviewmentioning

confidence: 99%