the planar cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/neuchatel.pdf ·...
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![Page 1: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/1.jpg)
The planar Cayley graphs are effectivelyenumerable
Agelos Georgakopoulos
Neuchatel, 20/10/15Agelos Georgakopoulos Planar Cayley graphs
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Groups need to act!
Let them act on the planeand be finitely generated
Agelos Georgakopoulos Planar Cayley graphs
![Page 3: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/3.jpg)
Groups need to act!
Let them act on the plane
and be finitely generated
Agelos Georgakopoulos Planar Cayley graphs
![Page 4: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/4.jpg)
Groups need to act!
Let them act on the planeand be finitely generated
Agelos Georgakopoulos Planar Cayley graphs
![Page 5: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/5.jpg)
Planar discontinuous groups
Planar discontinuous groups:= ‘discrete’ groups ofhomeomorphisms of S2, R2 or H2.discrete:= orbits have no accumulation points
Examples:
Agelos Georgakopoulos Planar Cayley graphs
![Page 6: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/6.jpg)
Planar discontinuous groups
Planar discontinuous groups:= ‘discrete’ groups ofhomeomorphisms of S2, R2 or H2.discrete:= orbits have no accumulation points
Examples:
Agelos Georgakopoulos Planar Cayley graphs
![Page 7: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/7.jpg)
Planar discontinuous groups
Planar discontinuous groups:= ‘discrete’ groups ofhomeomorphisms of S2, R2 or H2.discrete:= orbits have no accumulation points
Examples:
Agelos Georgakopoulos Planar Cayley graphs
![Page 8: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/8.jpg)
Known facts
Planar discontinuous groupsadmit planar Cayley graphsare virtually surface groupsadmit one-relator presentationsare effectively enumerable
see [Surfaces and Planar Discontinuous Groups, Zieschang, Vogt &Coldewey; Lecture Notes in Mathematics]or [Lyndon & Schupp].
Are Planar discontinuous groups exactly those having a planarCayley graph?
Definition: a group is planar, if it has a planar Cayley graph.
Agelos Georgakopoulos Planar Cayley graphs
![Page 9: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/9.jpg)
Known facts
Planar discontinuous groupsadmit planar Cayley graphsare virtually surface groupsadmit one-relator presentationsare effectively enumerable
see [Surfaces and Planar Discontinuous Groups, Zieschang, Vogt &Coldewey; Lecture Notes in Mathematics]or [Lyndon & Schupp].
Are Planar discontinuous groups exactly those having a planarCayley graph?
Definition: a group is planar, if it has a planar Cayley graph.
Agelos Georgakopoulos Planar Cayley graphs
![Page 10: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/10.jpg)
Known facts
Planar discontinuous groupsadmit planar Cayley graphsare virtually surface groupsadmit one-relator presentationsare effectively enumerable
see [Surfaces and Planar Discontinuous Groups, Zieschang, Vogt &Coldewey; Lecture Notes in Mathematics]or [Lyndon & Schupp].
Are Planar discontinuous groups exactly those having a planarCayley graph?
Definition: a group is planar, if it has a planar Cayley graph.
Agelos Georgakopoulos Planar Cayley graphs
![Page 11: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/11.jpg)
Known facts
Planar discontinuous groupsadmit planar Cayley graphsare virtually surface groupsadmit one-relator presentationsare effectively enumerable
see [Surfaces and Planar Discontinuous Groups, Zieschang, Vogt &Coldewey; Lecture Notes in Mathematics]or [Lyndon & Schupp].
Are Planar discontinuous groups exactly those having a planarCayley graph? NO!
Definition: a group is planar, if it has a planar Cayley graph.
Agelos Georgakopoulos Planar Cayley graphs
![Page 12: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/12.jpg)
Known facts
Planar discontinuous groupsadmit planar Cayley graphsare virtually surface groupsadmit one-relator presentationsare effectively enumerable
see [Surfaces and Planar Discontinuous Groups, Zieschang, Vogt &Coldewey; Lecture Notes in Mathematics]or [Lyndon & Schupp].
Are Planar discontinuous groups exactly those having a planarCayley graph? NO!
Definition: a group is planar, if it has a planar Cayley graph.
Agelos Georgakopoulos Planar Cayley graphs
![Page 13: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/13.jpg)
Charactisation of the finite planar groups
Definition: a group is planar, if it has a planar Cayley graph.
Theorem (Maschke 1886)
Every finite planar group is a group of isometries of S2.
Agelos Georgakopoulos Planar Cayley graphs
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The 1-ended planar groups
Theorem ((classic) Macbeath, Wilkie, ...)Every 1-ended planar Cayley graph correspondsto a group of isometries of R2 or H2.
see [Surfaces and Planar Discontinuous Groups, Zieschang, Vogt &Coldewey; Lecture Notes in Mathematics]quasi_6_4.gif (GIF Image, 560x420 pixels) http://www.math.cornell.edu/~mec/Winter2009/Mihai/section...
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Agelos Georgakopoulos Planar Cayley graphs
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The Cayley complex
Theorem (G ’12, Known?)A group has a flat Cayley complex if and only if ithas a accumulation-free Cayley graph.
(In which case it is a planar discontinuous group.)
Agelos Georgakopoulos Planar Cayley graphs
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Facial presentations
Theorem (G ’12)A Cayley graph admits an accumulation-free embeddingif and only if it admits a facial presentation.
based on...
Theorem (Whitney ’32)Let G be a 3-connected plane graph. Thenevery automorphism of G extends to ahomeomorphism of the sphere.
... in other words, every automorphism of G preserves facialpaths.
Agelos Georgakopoulos Planar Cayley graphs
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Facial presentations
Theorem (G ’12)A Cayley graph admits an accumulation-free embeddingif and only if it admits a facial presentation.
A facial presentation is a triple (P =< S | R >,σ, τ), whereσ is a spin, i.e. a cyclic ordering on S, andτ : S → {T ,F } decides which generators arespin-preserving or spin-reversing, so thatevery relator is a facial word.
based on...
Theorem (Whitney ’32)Let G be a 3-connected plane graph. Thenevery automorphism of G extends to ahomeomorphism of the sphere.
... in other words, every automorphism of G preserves facialpaths.
Agelos Georgakopoulos Planar Cayley graphs
![Page 18: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/18.jpg)
Facial presentations
Theorem (G ’12)A Cayley graph admits an accumulation-free embeddingif and only if it admits a facial presentation.
based on...
Theorem (Whitney ’32)Let G be a 3-connected plane graph. Thenevery automorphism of G extends to ahomeomorphism of the sphere.
... in other words, every automorphism of G preserves facialpaths.
Agelos Georgakopoulos Planar Cayley graphs
![Page 19: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/19.jpg)
Facial presentations
Theorem (G ’12)A Cayley graph admits an accumulation-free embeddingif and only if it admits a facial presentation.
A facial presentation is a triple (P =< S | R >,σ, τ), whereσ is a spin, i.e. a cyclic ordering on S, andτ : S → {T ,F } decides which generators arespin-preserving or spin-reversing, so thatevery relator is a facial word.
based on...
Theorem (Whitney ’32)Let G be a 3-connected plane graph. Thenevery automorphism of G extends to ahomeomorphism of the sphere.
... in other words, every automorphism of G preserves facialpaths.
Agelos Georgakopoulos Planar Cayley graphs
![Page 20: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/20.jpg)
planar Cayley graphs with accumulation points
Examples:
b
ac
Agelos Georgakopoulos Planar Cayley graphs
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planar Cayley graphs with accumulation points
Examples:
b
ac
Agelos Georgakopoulos Planar Cayley graphs
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planar Cayley graphs with accumulation points
Examples:
b
ac
Agelos Georgakopoulos Planar Cayley graphs
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planar Cayley graphs with accumulation points
Examples:
b
ac
Agelos Georgakopoulos Planar Cayley graphs
![Page 24: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/24.jpg)
planar Cayley graphs with accumulation points
Examples:
b
ac
Agelos Georgakopoulos Planar Cayley graphs
![Page 25: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/25.jpg)
What we didn’t know
Open Problems:
Problem (Mohar)
How can you splita planar Cayleygraph with > 1ends into simplerCayley graphs?
Problem (Droms et. al.)
Is there an effectiveenumeration of theplanar locally finiteCayley graphs?
Problem (Bonnington &Watkins (unpublished))
Does every planar3-connected locallyfinite transitive graphhave at least one facebounded by a cycle.
... and what about all the classical theory?
Agelos Georgakopoulos Planar Cayley graphs
![Page 26: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/26.jpg)
What we didn’t know
Open Problems:Problem (Mohar)
How can you splita planar Cayleygraph with > 1ends into simplerCayley graphs?
Problem (Droms et. al.)
Is there an effectiveenumeration of theplanar locally finiteCayley graphs?
Problem (Bonnington &Watkins (unpublished))
Does every planar3-connected locallyfinite transitive graphhave at least one facebounded by a cycle.
... and what about all the classical theory?
Agelos Georgakopoulos Planar Cayley graphs
![Page 27: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/27.jpg)
Dunwoody’s theorem
Theorem (Dunwoody ’09)If Γ is a group and G is a connected locally finite planar graphon which Γ acts freely so that Γ/G is finite, then Γ or an indextwo subgroup of Γ is the fundamental group of a graph ofgroups in which each vertex group is either a planardiscontinuous group or a free product of finitely many cyclicgroups and all edge groups are finite cyclic groups (possiblytrivial).
Agelos Georgakopoulos Planar Cayley graphs
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Classification of the cubic planar Cayley graphs
Theorem (G ’10, to appear in Memoirs AMS)Let G be a planar cubic Cayley graph. Then G iscolour-isomorphic to precisely one element of thelist.
Conversely, for every element of the list and anychoice of parameters, the corresponding Cayleygraph is planar.
Agelos Georgakopoulos Planar Cayley graphs
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Classification of the cubic planar Cayley graphs
Theorem (G ’10, to appear in Memoirs AMS)Let G be a planar cubic Cayley graph. Then G iscolour-isomorphic to precisely one element of thelist.Conversely, for every element of the list and anychoice of parameters, the corresponding Cayleygraph is planar.
Agelos Georgakopoulos Planar Cayley graphs
![Page 30: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/30.jpg)
Presentations of planar Cayley graphs withaccumulation points?
Recall that every accumulation-free Cayley graph has a facialpresentation.
What about
?Recall thatG has a facial presentation <=> G has a flat Cayley complex
How do we generalise?
Agelos Georgakopoulos Planar Cayley graphs
![Page 31: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/31.jpg)
Presentations of planar Cayley graphs withaccumulation points?
Recall that every accumulation-free Cayley graph has a facialpresentation.What about
?
Recall thatG has a facial presentation <=> G has a flat Cayley complex
How do we generalise?
Agelos Georgakopoulos Planar Cayley graphs
![Page 32: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/32.jpg)
Presentations of planar Cayley graphs withaccumulation points?
Recall that every accumulation-free Cayley graph has a facialpresentation.What about
?Recall thatG has a facial presentation <=> G has a flat Cayley complex
How do we generalise?
Agelos Georgakopoulos Planar Cayley graphs
![Page 33: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/33.jpg)
Presentations of planar Cayley graphs withaccumulation points?
Recall that every accumulation-free Cayley graph has a facialpresentation.What about
?Recall thatG has a facial presentation <=> G has a flat Cayley complex
How do we generalise?
Agelos Georgakopoulos Planar Cayley graphs
![Page 34: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/34.jpg)
Planar presentations
A presentation P =< S | R > is planar, if it can be endowed withspin data σ, τ so that
no two relator words crossevery relator contains an even number of spin-reversingletters.
σ is a spin, i.e. a cyclic ordering on Sτ : S → {T ,F } decides which generators are spin-preserving orspin-reversing
Agelos Georgakopoulos Planar Cayley graphs
![Page 35: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/35.jpg)
The Theorem
Theorem (G & Hamann, ’14)
A Cayley graph G is planar iff itadmits a planar presentation.
A presentation P =< S | R > is planar, if it can be endowed withspin data σ, τ so that
no two relator words crossevery relator contains an even number of spin-reversingletters.
The proof of forward direction involves ramifications ofDunwoody cuts. The proof of the backward direction iselementary, and mainly graph-theoretic, but hard.
Agelos Georgakopoulos Planar Cayley graphs
![Page 36: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/36.jpg)
The Theorem
Theorem (G & Hamann, ’14)
A Cayley graph G is planar iff itadmits a planar presentation.
A presentation P =< S | R > is planar, if it can be endowed withspin data σ, τ so that
no two relator words crossevery relator contains an even number of spin-reversingletters.
Cheat: this is a simplified definition, corresponding to the3-connected case;The general (2-connected) case is much harder to state and prove.
The proof of forward direction involves ramifications of Dunwoodycuts. The proof of the backward direction is elementary, and mainlygraph-theoretic, but hard.
Agelos Georgakopoulos Planar Cayley graphs
![Page 37: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/37.jpg)
The Theorem
Theorem (G & Hamann, ’14)
A Cayley graph G is planar iff itadmits a planar presentation.
A presentation P =< S | R > is planar, if it can be endowed withspin data σ, τ so that
no two relator words crossevery relator contains an even number of spin-reversingletters.
The proof of forward direction involves ramifications ofDunwoody cuts. The proof of the backward direction iselementary, and mainly graph-theoretic, but hard.
Agelos Georgakopoulos Planar Cayley graphs
![Page 38: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/38.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
Draw the corresponding tree TS := Cay < S | ∅ >accumulation-free in R2
Let D be a fundamental domain of TS w.r.t. N(R).We can choose D connected.
Note that if ∂D is nested, then G is planar
It remains to prove that ∂D is nested.
Agelos Georgakopoulos Planar Cayley graphs
![Page 39: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/39.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
Draw the corresponding tree TS := Cay < S | ∅ >accumulation-free in R2
Let D be a fundamental domain of TS w.r.t. N(R).We can choose D connected.
Note that if ∂D is nested, then G is planar
It remains to prove that ∂D is nested.
Agelos Georgakopoulos Planar Cayley graphs
![Page 40: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/40.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
Draw the corresponding tree TS := Cay < S | ∅ >accumulation-free in R2
Let D be a fundamental domain of TS w.r.t. N(R).We can choose D connected.
Note that if ∂D is nested, then G is planar
It remains to prove that ∂D is nested.
Agelos Georgakopoulos Planar Cayley graphs
![Page 41: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/41.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
Draw the corresponding tree TS := Cay < S | ∅ >accumulation-free in R2
Let D be a fundamental domain of TS w.r.t. N(R).We can choose D connected.
Note that if ∂D is nested, then G is planar
It remains to prove that ∂D is nested.
Agelos Georgakopoulos Planar Cayley graphs
![Page 42: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/42.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
Draw the corresponding tree TS := Cay < S | ∅ >accumulation-free in R2
Let D be a fundamental domain of TS w.r.t. N(R).We can choose D connected.
Note that if ∂D is nested, then G is planar
It remains to prove that ∂D is nested.
Agelos Georgakopoulos Planar Cayley graphs
![Page 43: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/43.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
Assume x , x ′ and y , y ′ is a non-nested pair
Observe that in R2 every cycle has two sides; a non-nested pairwould contradict this.
We’ll reverse engineer: given a cycle C in G, we want to definetwo ‘sides’ of C.
Two steps:—Step 1: if C comes from a relator W—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 44: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/44.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
Assume x , x ′ and y , y ′ is a non-nested pair
Observe that in R2 every cycle has two sides; a non-nested pairwould contradict this.
We’ll reverse engineer: given a cycle C in G, we want to definetwo ‘sides’ of C.
Two steps:—Step 1: if C comes from a relator W—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 45: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/45.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
Assume x , x ′ and y , y ′ is a non-nested pair
Observe that in R2 every cycle has two sides; a non-nested pairwould contradict this.
We’ll reverse engineer: given a cycle C in G, we want to definetwo ‘sides’ of C.
Two steps:—Step 1: if C comes from a relator W—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 46: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/46.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
Assume x , x ′ and y , y ′ is a non-nested pair
Observe that in R2 every cycle has two sides; a non-nested pairwould contradict this.
We’ll reverse engineer: given a cycle C in G, we want to definetwo ‘sides’ of C.
Two steps:—Step 1: if C comes from a relator W—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 47: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/47.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
Assume x , x ′ and y , y ′ is a non-nested pair
Observe that in R2 every cycle has two sides; a non-nested pairwould contradict this.
We’ll reverse engineer: given a cycle C in G, we want to definetwo ‘sides’ of C.
Two steps:—Step 1: if C comes from a relator W—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 48: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/48.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 1: if C comes from a relator W ∈ R
OK!
—Step 2: for general C, write C =∑
Wi , and apply Step 1 toeach Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 49: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/49.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 1: if C comes from a relator W ∈ R
OK!
—Step 2: for general C, write C =∑
Wi , and apply Step 1 toeach Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 50: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/50.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 1: if C comes from a relator W ∈ R
OK!
—Step 2: for general C, write C =∑
Wi , and apply Step 1 toeach Wi .
Agelos Georgakopoulos Planar Cayley graphs
![Page 51: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/51.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
We are inclined to say ‘let the inside of C be the union ofinsides of the Wi ’... but we don’t know what’s inside/outside!
Let’s still try:
IC := I14O24 . . . IkOC := O14I24 . . .Ok
Suppose it works; then anything works!
Agelos Georgakopoulos Planar Cayley graphs
![Page 52: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/52.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
We are inclined to say ‘let the inside of C be the union ofinsides of the Wi ’... but we don’t know what’s inside/outside!
Let’s still try:
IC := I14O24 . . . IkOC := O14I24 . . .Ok
Suppose it works; then anything works!
Agelos Georgakopoulos Planar Cayley graphs
![Page 53: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/53.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
We are inclined to say ‘let the inside of C be the union ofinsides of the Wi ’... but we don’t know what’s inside/outside!
Let’s still try:
IC := I14I24 . . . IkOC := O14O24 . . .Ok
Suppose it works;
IC := I14O24 . . . IkOC := O14I24 . . .Ok
Suppose it works; then anything works!
Agelos Georgakopoulos Planar Cayley graphs
![Page 54: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/54.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
We are inclined to say ‘let the inside of C be the union ofinsides of the Wi ’... but we don’t know what’s inside/outside!
Let’s still try:
IC := I14I24 . . . IkOC := O14O24 . . .Ok
Suppose it works;
IC := I14O24 . . . IkOC := O14I24 . . .Ok
Suppose it works; then anything works!
Agelos Georgakopoulos Planar Cayley graphs
![Page 55: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/55.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
We are inclined to say ‘let the inside of C be the union ofinsides of the Wi ’... but we don’t know what’s inside/outside!
Let’s still try:
IC := I14O24 . . . IkOC := O14I24 . . .Ok
Suppose it works;
then anything works!
Agelos Georgakopoulos Planar Cayley graphs
![Page 56: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/56.jpg)
Proof ideas: backward direction
If P =< S | R > is a planar presentation, then its Cayley graph Gis planar.
It remains to prove that ∂D is nested.
We want to define two ‘sides’ of C.—Step 2: for general C, write C =
∑Wi , and apply Step 1 to
each Wi .
We are inclined to say ‘let the inside of C be the union ofinsides of the Wi ’... but we don’t know what’s inside/outside!
Let’s still try:
IC := I14O24 . . . IkOC := O14I24 . . .Ok
Suppose it works; then anything works!Agelos Georgakopoulos Planar Cayley graphs
![Page 57: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/57.jpg)
The Theorem
Theorem (G & Hamann, ’14)
A Cayley graph G is planar iff itadmits a planar presentation.
CorollaryThe planar groups are effectively enumerable.
(Answering Droms et. al.)
Agelos Georgakopoulos Planar Cayley graphs
![Page 58: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/58.jpg)
Outlook
Generalise to include
Baumslag-Solitar_Cayley_3D.svg.png (PNG Image, 440 × 55... https://upload.wikimedia.org/wikipedia/commons/thumb/b/b...
1 of 1 15/09/15 11:50
Does anybody know if the groups having a Cayleycomplex embeddable in R3 have been characterised?
Agelos Georgakopoulos Planar Cayley graphs
![Page 59: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/59.jpg)
Outlook
Generalise to include
Baumslag-Solitar_Cayley_3D.svg.png (PNG Image, 440 × 55... https://upload.wikimedia.org/wikipedia/commons/thumb/b/b...
1 of 1 15/09/15 11:50
Does anybody know if the groups having a Cayleycomplex embeddable in R3 have been characterised?
Agelos Georgakopoulos Planar Cayley graphs
![Page 60: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/60.jpg)
Outlook
Theorem (Stallings ’71)Every group with >1 ends can be written as an HNN-extensionor an amalgamation product over a finite subgroup.
Can we generalise this to graphs?
Figure 3: A portion of the Cayley graph of HNN G A B !
the Cayley graph of G with respect to X . If X is any other finite generating set forG, then the number of ends of C G X is the same as that of C G X , and so werefer to this number as the number of ends of G. It is well-known that the number ofends of a finitely generated group is either 0 (in case G is finite), 1 (for example, ifG ), 2 (if G is virtually infinite cyclic) or uncountably infinite (if G is free ofrank two, for example.)
Stallings’s Theorem [15, 16] states that any finitely generated group with more thanone end is either a nontrivial free product with finite amalgamated subgroups, or anHNN extension with finite associated subgroups—that is, G is the fundamental groupof a graph of groups which has one edge, where the edge group is finite.
Given a subgraph f of a graph ", we define f to be the subgraph of " spanned bythe vertices which do not belong to f . We call f the complement of f . The set of edgeswhich belong neither to f nor to f (that is, the set of edges which have one endpointin f and the other in f ) is called the coboundary of f , and is denoted # f . Note that# f # f .
Let C be a Cayley graph for G, and suppose that G (and hence C ) has more thanone end. Then there is a cut in C : that is, an infinite connected subgraph e0 whosecomplement e0 is also connected and infinite, and whose coboundary is finite. LetE ge0 g G ge0 g G . Then there is an equivalence relation on E , definedas follows: given x and y in E , we set x y if x is, among elements of E , a maximalproper subset of y. (Note that x x for all x, and that if x y, then #x is disjoint fromy.) Let V denote the set of –classes.
Lemma 3.1 [5, 6] The sets V and E are, respectively, the vertex and edge sets of anundirected tree T , on which G acts. Each pair x x E represents the two possible
5
Agelos Georgakopoulos Planar Cayley graphs
![Page 61: The planar Cayley graphs are effectively enumerablehomepages.warwick.ac.uk/~maslar/Neuchatel.pdf · 2015. 10. 20. · Known facts Planar discontinuous groups admit planar Cayley graphs](https://reader035.vdocuments.net/reader035/viewer/2022063011/5fc59ce005eb4124333f6109/html5/thumbnails/61.jpg)
Thank you!
Agelos Georgakopoulos Planar Cayley graphs