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).