Tag: measured lamination

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~!

A report from the Workshop in Geometric Topology @ Utah (part 1)

Conan9 Comments

I went to Park City this passed week for the Workshop in Geometric Topology. It was a quite cool place filled with ski-equipment stores, Christmas souvenir shops, galleries and little wooden houses for family winter vacations. Well, as you may have guessed, the place would look very interesting in summer. :-P

As the ‘principal speaker’, Professor Gabai gave three consecutive lectures on his ending lamination space paper (this paper was also mentioned in my last post). I would like to sketch some little pieces of ideas presented in perhaps couple of posts.

Classification of simple closed curves on surfaces

Let S_{g,p} denote the (hyperbolic) surface of genus g and p punchers. There is a unique geodesic loop in each homotopy class. However, given a geodesic loop drew on the surface, how would you describe it to a friend over telephone?

Here we wish to find a canonical way to describe homotopy classes of curves on surfaces. This classical result was originally due to Dehn (unpublished), but discovered independently by Thurston in 1976. For simplicity let’s assume for now that S is a closed surface of genus g.

Fix pants decomposition \mathcal{T} of S, \mathcal{T} = \{ \tau_1, \tau_2, \cdots, \tau_{3g-3} \} is a disjoint union of 3g-3 ‘cuffs’.

As we can see, any simple closed curve will have an (homology) intersection number with each of the cuffs. Those numbers are non-negative integers:

Around each cuff we may assign an integer twist number, for a cuff with intersection number n and twist number z, we ‘twist’ the curve inside a little neighborhood of the cuff so that all transversal segments to the cuff will have z intersections with the curve.

Negative twists merely corresponds to twisting in the other direction:

Theorem: Every simple closed curve is uniquely defined by its intersection number and twisting number w.r.t each of the cuffs.

Conversely, if we consider multi-curves (disjoint union of finitely many simple closed curves) then any element in \mathbb{Z}^{3g-3} \times \mathbb{Z}_{\geq 0}^{3g-3} describes a unique multi-curve.

To see this we first assume that the pants decomposition comes with a canonical ‘untwisted’ curve connecting each pairs of cuffs in each pants. (i.e. there is no god given ‘0’ twist curves, hence we have to fix which ones to start with.)

In the example above our curve was homotopic to the curve ((1,2), (2,1), (1,-4)).

In other words, pants decompositions (together with the associated 0-twist arcs) give a natural coordinate chart to the set of homotopy class of (multi) curves on a surface. i.e. they are perimetrized by \mathbb{Z}^{3g-3} \times \mathbb{Z}_{\geq 0}^{3g-3}.

For the converse, we see that any triple of integers can be realized by filling the pants with a unique set of untwisted arcs:

In fact, this kind of parametrization can be generalized from integers to real numbers, in which case we have measured laminations instead of multi-curves and maximal train trucks on each pants instead of canonical untwisted arcs. i.e.

Theorem: (Thurston) The space of measured laminations \mathcal{ML}(S) on a surface S of genus g is parametrized by \mathbb{R}^{3g-3} \times \mathbb{R}_{\geq 0}^{3g-3}. Furthermore, the correspondence is a homeomorphism.

Here the intersection numbers with the cuffs are wrights of the branches of the train track, hence it can be any non-negative real number. The twisting number is now defined on a continuous family of arcs, hence can be any real number, as shown below:

As we can see, just as in the case of multi-curves, any triple of real numbers assigned to the cuffs can be realized as the weights of branches of a train track on the pants.