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Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
The Action of SL2(Z) on OrigamisA subset of the Quadratic Differentials
Paige Helms
(In collaboration with T. Aougab, Z. Cui, A. Gary, T. Kim, J. Rustad)
University of California, Riverside
January 26th, 2019
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Outline
• Curves on Surfaces
• Origami
• SL2(Z) Action
• The Invariant and Our Results
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Curves on Surfaces
We will consider curves that are:
• Simple, Closed, Essential, Oriented
• In Pairs: (λ, µ)
• Minimally Intersecting: min(λ ∩ µ)
• Filling: Σg \ (λ ∪ µ) ' D2
Examples
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Minimal Intersection
Lemma
Suppose (α, β) is a pair of curves which fill Σg , g ≥ 1. Theni(α, β) = 2g − 1. Aougab-Haung 2013
Sketch of the proof:
• Let i(α, β) denote the intersection number of curves (α, β)
• χ(X ) = 2− 2g = Vertices - Edges + Faces
• Since Vertices = i(α, β) and Edges = 2i(α, β),
• 2− 2g = i(α, β)− 2i(α, β)+ Faces
• 2− 2g = −i(α, β) + Faces ≥ −i(α, β) + 1
• i(α, β) ≥ 2g − 1.
• Since Σg \ (α ∪ β) is a single disk, i(α, β) = 2g − 1.
Relabel n := 2g − 1.
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Surfaces
Examples of Surfaces
If we take the collection of all surfaces up to homeomorphism andquotient the set by conformal maps between surfaces, we get theTeichmuller space, Tg .
The cotangent bundle of Tg is naturally identified with the spaceof quadratic differentials QDg .
Origamis are a subset of QDg .
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Origami
An origami is a surface obtained from gluing up the boundary of aregion in C that is tiled by congruent squares.
Origamis can also be naturally identified as a branched cover of asquare torus, where there is a single branch point.
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
SL2(Z) Action
SL2(R) has a natural action on QDg :SL2(R)×QDg → QDg
This action restricts to SL2(Z) on the set of origamis:SL2(Z)× {Origamis} → {Origamis}
Take as a basis (S =
[1 10 1
],R =
[0 −11 0
]) for SL2(Z).
Then the action can be described in terms of the basis as follows:
R · (λ, µ) = (µ, λ−1) S · (λ, µ) = (λ, µ · λ−1)
This action of SL2(Z) partitions the set of all pairs (λ, µ) intoorbits.
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
SL2(Z) Action
R · (λ, µ) = (µ, λ−1) S · (λ, µ) = (λ, µ · λ−1)
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Monodromy Explanation
Because an origami is a branched cover of T 2, there is anassociated monodromy representation of the π1(T 2 \ {pt}) = F2.
The image of this representation is a subgroup of Sn, with n thenumber of squares, called the monodromy group.
As a consequence of n being odd, the monodromy group of thecover is ≤ An.
It is known that the monodromy group serves as an invariant ofthe orbits under the SL2(Z) action on the set of origamis.
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
A Lower Bound
Main Question
For any g , how many orbits exist under this SL2(Z) action?
Under this invariant, we expect that ∀Σg with g > 4, there are atleast two orbits under the SL2(Z) action.
Our Results So Far
We have shown this to be true for g = 5, 6, 7.
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Our Results
Table 1: Orbit Data
Genus n = 2g − 1 Orbits Monodromy Group(s)
g=3 5 1 A5
g=4 7 4 A7
g=5 9 11 A9, L2(8), L2(8) o Z3
g=6 11 ≥ 2 A11, M11
g=7 13 ≥ 2 A13, SL3(3)
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
Acknowledgements
Thanks to:
21st Annual NCUWM, 2019
University of California, Riverside
ICERM and my collaborators
National Science Foundation
Curves on Surfaces Origami SL2(Z) Action The Invariant and Our Results Acknowledgements
References
T. Aougab, B. Menasco, M. Nieland. Square-tiled Surfaceswith Connected Leaves in the Minimal Stratum, in preparation.
T. Aougab, S. Huang , Minimally Intersecting Curves onSurfaces, arXiv:1312.0913v2 (2013).
G. Schmithusen, An Algorithm for Finding the Veech Group ofan Origami, arXiv:math/0401185v1 (2004).
D. Zmiakou, Origamis and permutation groups, Ph.D. thesis,Universite Paris-Sud, Orsay (2011).
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