Month: April 2012

A train track on twice punctured torus

Conan4 Comments

This is a non-technical post about how I started off trying to prove a lemma and ended up painting this:

One of my favorite books of all time is Thurston‘s ‘Geometry and Topology of 3-manifolds‘ (and I just can’t resist to add here, Thurston, who happen to be my academic grandfather, is in my taste simply the coolest mathematician on earth!) Anyways, for those of you who aren’t topologists, the book is online and I have also blogged about bits and parts of it in some old posts such as this one.

I still vividly remember the time I got my hands on that book for the first time (in fact I had the rare privilege of reading it from an original physical copy of this never-actually-published book, it was a copy on Amie‘s bookshelf, which she ‘robbed’ from Benson Farb, who got it from being a student of Thurston’s here at Princeton years ago). Anyways, the book was darn exciting and inspiring; not only in its wonderful rich mathematical content but also in its humorous, unserious attitude — the book is, in my opinion, not an general-audience expository book, but yet it reads as if one is playing around just to find out how things work, much like what kids do.

To give a taste of what I’m talking about, one of the tiny details which totally caught my heart is this page (I can’t help smiling each time when flipping through the book and seeing the page, and oh it still haunts me >.<):

This was from the chapter about Kleinian groups, when the term ‘train-track’ was first defined, he drew this image of a train(!) on moving on the train tracks, even have smoke steaming out of the engine:

To me such things are simply hilarious (in the most delightful way).

Many years passed and I actually got a bit more into this lamination and train track business. When Dave asked me to ‘draw your favorite maximal train track and test your tube lemma for non-uniquely ergodic laminations’ last week, I ended up drawing:

Here it is, a picture of my favorite maximal train track, on the twice punctured torus~! (Click for larger image)

Indeed, the train is coming with steam~

Since we are at it, let me say a few words about what train tracks are and what they are good for:

A train track (on a surface) is, just as one might expect, a bunch of branches (line segments) with ‘switches’, i.e. whenever multiple branches meet, they must all be tangent at the intersecting point, with at least one branch in each of the two directions. By slightly moving the switches along the track it’s easy to see that generic train track has only switches with one branch on one side and two branches on the other.

On a hyperbolic surface S_{g,p}, a train track is maximal if its completementry region is a disjoint union of triangles and once punctured monogons. i.e. if we try to add more branches to a maximal track, the new branch will be redundant in the sense that it’s merely a translate of some existing branch.

As briefly mentioned in this post, train tracks give natural coordinate system for laminations just like counting how many times a closed geodesic intersect a pair of pants decomposition. To be slightly more precise, any lamination can be pushed into some maximal train track (although not unique), once it’s in the track, any laminations that’s Hausdorff close to it can be pushed into the same track. Hence given a maximal train track, the set of all measured laminations carried by the train track form an open set in the lamination space, (with some work) we can see that as measured lamination they are uniquely determined by the transversal measure at each branch of the track. Hence giving a coordinate system on \mathcal{ML})(S).

Different maximal tracks are of course them pasted together along non-maximal tracks which parametrize a subspace of \mathcal{ML}(S) of lower dimension.

To know more about train tracks and laminations, I highly recommend going through the second part of Chapter 8 of Thurston’s book. I also mentioned them for giving coordinate system on the measured lamination space in the last post.

In any case I shall stop getting into the topology now, otherwise it may seem like the post is here to give exposition to the subject while it’s actually here to remind myself of never losing the Thurston type childlike wonder and imagination (which I found strikingly larking in contemporary practice of mathematics).

Filling and unfilling measured laminations

Conan2 Comments

(images are gradually being inserted ~)

I’m temporarily back into mathematics to (try) finish up some stuff about laminations. While I’m on this, I figured maybe sorting out some very basic (and cool) things in a little post here would be a good idea. Browsing through the blog I also realized that as a student of Dave’s I have been writing surprisingly few posts related to what we do. (Don’t worry, like all other posts in this blog, I’ll only put in stuff anyone can read and hopefully won’t be bored reading :-P)

Here we go. As mentioned in this previous post, my wonderful advisor has proved that the ending lamination space is connected and locally connected (see Gabai’08).

Definition: Let S_{g,p} be a hyperbolic surface of genus g and p punctures. A (geodesic) lamination L \subseteq S is a closed set that can be written as a disjoint union of geodesics. i.e. L = \sqcup_{\alpha \in I} \gamma_\alpha where each \gamma_\alpha is a (not necessary closed) geodesic, \gamma is called a leaf of L.

Let’s try to think of some examples:

i) One simple closed geodesic

ii) A set of disjoint simple closed geodesics

iii) A non-closed geodesic spirals onto two closed ones

iV) Closure of a single simple geodesic where transversal cross-sections are Cantor-sets

An ending lamination is a lamination where
a) the completement S \backslash L is a disjoint union of discs and once punctured discs (filling)
b) all leaves are dense in L. (minimal)

Exercise: example i) satisfies b) and example iv) as shown satisfies both a) and b) hence is the only ending lamination.

It’s often more natural to look at measured laminations, for example as we have seen in the older post, measured laminations are natural generalizations of multi-curves and the space \mathcal{ML}(S) is homeomorphic to \mathbb{R}^{6g-6+2p} (Thurston) with very natural coordinate charts (given by train-tracks).

Obviously not all measured laminations are supported on ending laminations (e.g. example i) and ii) with atomic measure on the closed curves.) It is well known that if a lamination fully supports an invariant measure, then as long as the base lamination satisfies a), it automatically satisfies b) and hence is an ending lamination. This essentially follows from the fact that having a fully supported invariant measure and being not minimal implies the lamination is not connected and hence won’t be filling.

Exercise:Example iii) does not fully support invariant measures.

Scaling of the same measure won’t effect the base lamination, hence we may eliminate a dimension by quotient that out and consider the space of projective measured laminations \mathcal{PML}(S) \approx \mathbb{S}^{6g-7+2p}. Hence we may decompose measured laminations into filling and unfilling ones. i.e.

\mathcal{PML}(S) = \mathcal{FPML}(S) \sqcup \mathcal{UPML}(S)

where \mathcal{FPML}(S) projects to the ending laminations via the forgetting measure map \pi.

This decomposition of the standard sphere \mathbb{S}^{6g-7+2p} is mysterious and very curious in my opinion. To get a sense of this, let’s take a look at the following facts:

Fact 1: \mathcal{UPML} is a union of countably many disjoint hyper-discs (i.e. discs of co-dimension 1).

Well, if a measured lamination is unfilling, it must contain some simple closed geodesic as a leaf (or miss some simple closed geodesic). For each such geodesic C, there are two possible cases:

Case 1: C is non-separating. The set of measured laminations that missed C is precisely the set of projective measured laminations supported on S_{g-1, p+2}, hence homeomorphic to \mathbb{S}^{6g-13+2p+4} = \mathbb{S}^{(6g-7+2p)-2} we may take any such measured lamination, disjoint union with C, we may assign any ratio of wrights to C and the lamination. This corresponds to taking the cone of \mathbb{S}^{(6g-7+2p)-2} with vertex being the atomic measure on C. Yields a disc of dimension (6g-7+2p)-1.

Case 2: C is separating. Similarly, the set of measured laminations missing C is supported on two connected surfaces with total genus g and total punctures p+2.

To describe the set of projective measured laminations missing C, we first determine the ratio of measure between two connected components and then compute the set of laminations supported in each component. i.e. it’s homeomorphic to [0,1] \times \mathbb{S}^{d_1} \times \mathbb{S}^{d_2}/\sim where d_1+d_2 = 6g-2*7+2(p+2) = 6g-10+2p and (0, x_1, y) \sim (0, x_2, y) and (1, x, y_1) \sim (1, x, y_2).

Exercise: check this is a sphere. hint: if d_1 =d_2 = 1, we have:

Again we cone w.r.t. the atomic measure corresponding to C, get a hyper disc.

At this point you may think ‘AH! \mathcal{UPML} is only a countable union of hyper-discs! How complicated can it be?!’ Turns out it could be, and (unfortunately?) is, quite messy:

Fact 2: \mathcal{UPML} is dense in \mathcal{PML}.

This is easy to see since any filling lamination is minimal, hence all leaves are dense, we may simply take a long segment of some leaf where the beginning and end point are close together on some transversal, close up the segment by adding a small arc on the transversal, we get a simple closed geodesic that’s arbitrarily close to the filling lamination in \mathcal{PML}. Hence the set of simple closed curves with atomic measure are dense, obviously implying \mathcal{UPML} dense.

So how exactly does this decomposition look like? I found it very mysterious indeed. One way to look at this decomposition is: we know two \mathcal{UPML} discs can intersect if and only if their corresponding curved are disjoint. Hence in some sense the configuration captures the structure of the curve complex. Since we know the curve complex is connected, we may start from any disc, take all discs which intersect it, then take all discs intersecting one of the discs already in the set, etc.

We shall also note that all discs intersecting a given disc must pass through the point corresponding to the curve at the center. Hence the result will be some kind of fractal-ish intersecting discs:

(image)

Yet somehow it manages to ‘fill’ the whole sphere!

Hopefully I have convinced you via the above that countably many discs in a sphere can be complicated, not only in pathological examples but they appear in ‘real’ life! Anyways, with Dave’s wonderful guidance I’ve been looking into proving some stuff about this (in particular, topology of \mathcal{FPML}). Hopefully the mysteries would become a little clearer over time~!