Hyperbolic geometry and manifolds: Arithmetic manifolds

Embedded totally geodesic submanifolds in small volume hyperbolic manifolds- Michelle Chu (University of Illinois, Chicago)

Abstract: The smallest volume cusped hyperbolic 3-manifolds are arithmetic and contain many immersed but not embedded closed totally geodesic surfaces. In this talk we discuss nonexistence of codimension-1 closed embedded totally geodesic submanifolds in small volume hyperbolic manifolds of higher dimensions. This is joint work with Long and Reid.

Arithmetic manifolds and their geodesic submanifolds- Nicholas Miller (University of California Berkeley)

Abstract: It is a well known consequence of the Margulis dichotomy that when an arithmetic hyperbolic manifold contains one totally geodesic hypersurface, it contains infinitely many. Both Reid and McMullen have asked conversely whether the existence of infinitely many geodesic hypersurfaces implies arithmeticity of the corresponding hyperbolic manifold. In this talk, I will discuss recent results answering this question in the affirmative. In particular, I will describe how this follows from a general superrigidity style theorem for certain natural representations of fundamental groups of hyperbolic manifolds. I will also discuss a recent extension of these techniques into the complex hyperbolic setting, which requires the aforementioned superrigidity theorems as well as some theorems in incidence geometry. This is joint work with Bader, Fisher, and Stover.

Deligne-Mostow lattices and line arrangements in complex projective 2-space- Irene Pasquinelli (Institut de Mathématiques de Jussieu, Paris)

Abstract: In 1983, Hirzebruch considers some arrangements of complex lines in complex projective 2-space. Then he shows that a suitable branched cover ramified along the line arrangement is a complex hyperbolic manifold. This manifold turns out to be one of the well known Deligne-Mostow lattices. In the first part of this talk I will introduce you to the complex hyperbolic space, to its group of isometries and to the Deligne-Mostow lattices. Then I will tell you about Hirebruch's construction. In the second part, I will first explain how Hirzebruch's work has been generalised by Bartel, himself and Hoefer to all of the Deligne-Mostow lattices. Then I will tell you how, in a joint work with Elisha Falbel, we created an explicit dictionary between line arrangements and fundamental domains for the lattices. I will also explain one of the applications of this. In fact, we use this result to contribute to the problem of creating a complex analogue to the hybridisation construction.


10 replies on “Hyperbolic geometry and manifolds: Arithmetic manifolds”

Hi everyone! Comments or questions are always welcomed. I also invite you to chat with me on Zoom during my office hour on Friday June 5!

I am Irene Pasquinelli, currently a postdoc in Paris.
Being my first virtual conference, I welcome any feedback, suggestions or comments about possible improvements to the presentation, as well as any maths question you might have. Feel free to ask here (now I will be notified!) or during my office hours!

Is it known if the sole non-arithmetic DM lattice in PU(3,1) also has a nice description arising from a hyperplane arrangement?

Not that I know of, unfortunately.
Showing that the branched cover has a complex hyperbolic structure uses Miyaoka Yau which, I think, has a (slightly more complicated) higher dimensional equivalent. So it might be possible to explore that. But I guess the first step would be to think branched covers of… what?

Hi everyone! Thanks for joining our session on arithmetic manifolds. If you have any questions or comments feel free to drop by my office hours on Tuesday (6/2) at 2 EST or send me a message.

Hey Michelle,

Where did the motivation for this conjecture come from (if not just it’s intrinsic interest as a way of saying that hyperbolic cusped n-manifolds of smallest volume are not like surfaces)? Are there related questions that seem tractable/interesting? If you look at the next smallest volume for cusped hyperbolic 3 and 4 manifolds , what can be said about embedded, closed, tg. hypersurfaces? What about tg. immersions with self intersection at most some number etc. ? Answer however much you’d like/ not like.

Hi James,

I think the main motivation for the conjecture is what happens in dimensions 3 and 5+.

In dimension 3, the next smallest cusped hyperbolic manifolds are m006, m007 which are not arithmetic, followed by m009 and m010 which are arithmetic. I don’t know what happens in these 4 cases, but one could at least check Hakenness with Regina.
In dimension 4, we give an example in the paper arXiv:2005.11256 having Euler characteristic 2 (next smallest volume) that does not have any closed embedded tg 3-manifolds. Unfortunately we cannot say much more at this time. Like I mentioned in the talk, our results only apply to 22 explicit examples, but there are other smallest volume hyperbolic 4 manifolds out there.

There are many related questions, some of which you asked and some listed at the end of the paper. Some other questions: how many commensurability classes exist containing smallest volume hyperbolic 4-manifolds? What is the smallest volume 4-manifold that contains a closed embedded tg hypersurface? Can you identify manifolds which contain exactly k many closed embedded tg hypersurfaces? Can you identify manifolds, necessarily nonarithmetic, which contain exactly k many immersed tg hypersurfaces? We know very little about hyperbolic 4-manifolds, including what the smallest possible volume for a closed hyp 4-manifold.

I am happy to chat more!

Hi Michelle,

I was actually wondering about the first question you listed but haven’t had any time to think about it. I was wondering if one can use Prasad volume formula here to produce a finite list of possible commensurability classes that can support an orbifold of volume <= to the minimal volume manifold and then try to analyze which of these commensurability classes support a torsion free lattice of minimal volume. I have no idea how feasible this is but would love to hear your thoughts on it!

Hi Nick,

There are results by Belolipetsky and others where they use Prasad’s volume formula find small volume 4-orbifolds. However, somehow it seems difficult to go from orbifolds to manifolds. I have not thought about it much myself though.

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