# Seifert-Fibered 3-Manifolds and Rotation Numbers

I was trying to understand how the "ziggurats" come about in the paper by Calegari and Walker.

Motivating Question Given a free group F, and an element w of F, and given values of the rotation numbers of the generators, what is the set of possible rotation numbers of w?

I think it a good question is which of the "rationality conjectures" remain open at this time.

• Let $w$ be arbitrary and $r,s \in \mathbb{Q}$. Then $R(w,r-,s-)\in \mathbb{Q}$
• Let $w$ be arbitrary and $r,s \in \mathbb{Q}$. Then $R(w,r,s)\;\;\;\;\in \mathbb{Q}$

Here, $R(w,r-,s-)$ is the supremum of the rotation numbers of $w$ over a certain set of maps.
They prove if $r, s \in \mathbb{Q}$ then $R(w,r,s) \in \mathbb{Q}$ this is their rationality theorem. They restrict to positive words "w" in the free group on two generators, $F_2$.

I got lost with terms like "bounded cohomology" and "Seifert-Fibered 3-manifolds" and I really couldn't piece together enough to reproduce their graphics. I know what a rotation number is:

$$\omega(f) = \lim_{n \to \infty} \frac{F^n(x) - x}{n}$$

This is a cool enough problem for me, but Calegori seems to have been motivated by broader questions in Topology. The character variety of $\text{Homeo}^+(S^1)$ and the rotation number is a representation of the free group into the real numbers $\text{rot}:F_2 \to \mathbb{R}/\mathbb{Z}$. Even if you fixed the word $w$, there seem to be could be many possible representatives so that $X(w,r,s)$ is the set of possible rotation numbers and $R(w,r,s)$ is the supremum of that.

While this could be common knowledge to geometers, this doesn't look like a 3-manifold classification problem at all. If there's a circle bundle over a surface, then maybe as we continuously move around the surface the circle maps to itself.

This was found doing simple reverse-lookup on Google Scholar. There could be no relation at all. One paper emphasizes Seifert fiberbered 3-manifolds, the other mentions dynamical systems.

• It sounds like what you want is broad background to the Calegari--Walker paper and related work. I recommend Calegari's two books: Foliations and the Geometry of 3-Manifolds and scl. Both can be downloaded in pdf format from his website: math.uchicago.edu/~dannyc . – HJRW Oct 26 '17 at 8:57
• By the way, you're right that the connection between representations of groups into $\mathrm{Homeo}(S^1)$ and 3-manifold topology is via circle bundles. Seifert-fibred 3-manifolds are a kind of generalization of circle bundles over surfaces. – HJRW Oct 26 '17 at 8:59