# Colin J. Fan - DIMACS REU 2020

 Email: colin dot fan at rutgers dot edu Office: Hill Center 323 Group: Number Theory and Hyperbolic Geometry Mentor: Alex Kontorovich Home Institution: Rutgers University Department of Mathematics

Colleague Pages: Brittany Gelb, Brandon Gomes, Saket Shah, and Kai Shaikh.

### Project: The Ehrenpreis Conjecture for Punctures

Motivation: The uniformization theorem states that any connected Riemann surface has universal covering that is biholomorphic to either the Riemann sphere, the complex plane, or the upper half plane. If the genus of the surface is at least two, then the universal cover is the upper half plane. It follows that, any two Riemann surfaces of genus at least two then have covers (not necessarily finite) that are identical, and therefore "close in geometry." What can be said about the existence of finite covers that are "close in geometry?"

The Ehrenpreis Conjecture roughly states that for any two compact Riemann surfaces without boundary of genus at least two, and any angle distortion factor, there exists two finite degree covers for each surface, such that these covers are "close in geometry." That is, there exists a homeomorphism between the covers which only distort angles by at most the distortion factor. This conjecture was first proven by Jeremy Kahn and Vladimir Markovic in 2011.

What can be said about Riemann surfaces with punctures/cusps? The long term goal of this project is to prove the Ehrenpreis conjecture for cusped Riemann surfaces. Step one is to collect examples of finite covers of cusped surfaces, such as the fundamental domain of the upper half plane under the modular group and fractional linear action. Here is the fundamental domain and its orbits in question (T is translation to the right, and S is the reciprocal rotated by π):

This project will be tackled along with Saket Shah, and Kai Shaikh.

This work is supported by the Rutgers Department of Mathematics and the National Science Foundation Grant DMS-1802119.

### Week 1: May 26 - May 31

This week was concerned with deciding what we will be studying this summer. We were introduced to four projects: "Lean and Computer Assisted Proofs," "Classification of Integral Polyhedra," "Reflection Groups in the Hyperbolic Disk," and "The Ehrenpreis Conjecture for Punctured Surfaces." Professor Kontorovich gave us introductions to all four projects, and Professor Jeremy Kahn also guest lectured on the last project. On Friday, we had all decided which projects we would pursue for the summer. See Brittany and Brandon's page above to see what they will be working on!

Brandon also helped me make this website and guided me through my first Python experiment.

Two fun facts I learned this week: The fundamental domain in the hyperbolic disk looks like a triangle with one vertex at the boundary, and two other vertices colinear with the origin such that the geodesic connecting the boundary vertex and the other two vertices form angles of π/3 with the straight line connecting those two vertices. So the sides of the the triangle are two arcs of circles, and one straight line. Next, the area of the fundamental domain is π/3 with respect to the volume form generated by the Poincaré metric. This is a nice exercise in calculus 3!

### Week 2: June 1 - June 7

To start the project off, we read the first lecture of Professor Kontorovich's notes that contained exposition on closed geodesics in the fundamental domain and its relations to the modular group and number theory. It was extremely enlightening to see a purely geometric object (geodesics) be modeled in the language of algebra and number theory. To work towards the goal of actually constructing a cover for the fundamental domain, our group worked on converting many ideas of the notes into Python code. We got as far as being able to draw closed geodesics in the fundamental domain given any hyperbolic matrix!

Fun fact I learned this week while reading the above notes (Iwasawa decomposition): Any element in the 2 × 2 special linear group over the reals can be decomposed uniquely as the product of three types of matrices: A diagonal matrix where the entries are positive and inverse to each other, a shear matrix, and a rotation matrix. This fact is useful for showing the correspondence between the hyperbolic unit tangent bundle (and therefore geodesics!), and the 2 × 2 projective special linear group over the reals. The decomposition follows from Gram-Schmidt orthogonalization. See Keith Conrad's notes for a proof of the fact as well as notes on its relevance to hyperbolic geometry (page 8)!

### Week 3: June 8 - June 14

Last week we converted the language of geometry into the language of linear algebra and number theory via hyperbolic matrices, quadratic forms, and continued fraction expansions. This week, we followed similarly and converted closed geodesics into combinatorial information via cutting sequences. In short, by triangulating the hyperbolic plane via Farey sequences, we were able to track the movement of closed geodesics as to how it passes through each triangle. Such treatment can be found in Series's paper, "The Modular Surface and Continued Fractions." We also met with Professor Kahn to talk about cutting sequences, and how it allows us to immerse pants into the modular surface. In Python, given a closed geodesic, we were able to convert it into a continued fraction, and from this, convert to a cutting sequence.

The reason why we care about cutting sequences, is because we want a way to not only immerse pair of pants into the modular surface, but to also immerse good pair of pants. To start off, a pair of pants when viewed in the plane can be seen as a two-holed disk without boundary, where the boundary (with an orientation) are the cuffs of the pants. One can then draw a theta within this disk that contains both holes, separated by the straight line of the theta. We are able to add an orientation to this theta, and the immersion of the pants is one that is modeled by the immersion of this theta. When mapped nicely enough, this theta graph allows us to track its behavior via cutting sequences, and if the cutting sequences of the theta graph are "compatible," then we have a way to immerse a pair of pants. It is important that we did this work for geodesics already, as we want our pants to be "good." That is, their cuffs are geodesics. In Python, given a range of traces (equivalent to length of closed geodesic), we were able to obtain geodesics with traces in that range, convert that information into cutting sequences, and see which sequences were compatible with one another. This allowed us to output triples of compatible geodesics that represented good pants being immersed in the modular surface. Here are some pictures:

Why do we care about pants? Well one day we want to construct finite covers for surfaces, and the proposed attack is to construct covers via good pants!

Two fun facts I learned this week from Series's paper and meetings with Professor Kontorovich: The mapping of geodesics to cutting sequences is in some way, injective. That is, if two geodesics have the same cutting sequence, then they coincide. Moreover, given a curve that admits a cutting sequence, its cutting sequence is invariant under tightening the curve to become a geodesic. That is, if a curve connects two points, then the geodesic that connects those two points has the same cutting sequence as the original curve.

### Week 4: June 15 - June 21

Using the combinatorial information from last week via cutting sequences, we were able to "close up" the modular surface using finitely many pants and finitely many "eyes!" Note that we cannot use finitely many pants to cover the fundamental domain, as each pants is bounded, while the fundamental domain is unbounded. Thus, we need another type of object that is compatible with pants to help close up the fundamental domain. This object is a degenerate pants, called an "eye" that is modeled by one closed geodesic, and one "cusped" geodesic that is not closed, and goes to infinity in the modular surface. This cusped geodesic plays the role of helping to close up the fundamental domain once we run out of pants. Moreover, we can make it compatible with cutting sequences as an eye is modeled by geodesics, and therefore convert the language of eyes to the language of combinatorics as we had done for pants last week.

We were then able to convert the problem of gluing together pants and eyes into a problem of finding a certain vector in the null space of a matrix consisting of ±1 and 0. This vector must be nonzero, and all entries must be nonnegative integers. With the help of PuLP and Python, given a collection of pants and eyes from a range of traces, we could convert this information into a matrix, and check for a proper solution. We found that the pants and eyes generated from the range of traces between 10 and 30 closed up the fundamental domain as the converted matrix contained a good vector in its null space. This closing up consisted of two pants and two eyes!

### Week 5: June 22 - June 28

Given a range of traces, it would be nice to find a collection of pants and eyes that have cuffs with trace in that range that both close up the fundamental domain and have a nice distribution. To do this, we made the following assumptions: not only will we find a proper solution vector for the matrix obtained from pants and eyes, but we will extend it to allow nonnegative solutions in the reals and minimize its length with respect to the $\ell^2$-norm. The first assumption was made to ease computing time, as finding a solution in the integers was quite difficult for the computer. Note that such a real solution allows us to find a nearby rational solution, as the null space is generated by vectors with rational entries, and therefore we can find a proper integer solution. To find a minimal solution in $\ell^2$, we used CVXPY.

We were expecting to get an even distribution of pants and eyes, however in every case we tested, this was not the case. We then tested other $\ell^p$-norms and weights on the solution vector to see if we could find a balanced distribution. In every case, we noticed that lots of pants and eyes were getting unused when closing up the surface, and that while the number of pants had low variance between one another, the number of eyes used varied highly.

### Week 6: June 29 - July 5

Despite not being able to close up the surface in a "balanced" fashion, we moved on to attempting to find an explicit cover for the modular surface. Step one is to match the cuff of pants together.

### Week 7: July 6 - July 12

Once we match cuffs together, we need to care about the geometry, and the twisting of the pants, as we need to control the final geometry of the covering space. This twisting is measured by "shears."