**The structure theorem for train track splitting sequences**

joint work with Howard Masur and Lee Mosher. Related to recent work of Bestvina and Feighn on folding sequences in the .

**1. Introduction **

Consider surface of genus and boundary components.

There are several spaces on which mapping class group of act. We want to understand the coarse geometry of these spaces.

**2. Arcs and curves **

A curve is a proper embedding of into . An arc is a proper embedding of interval into . Say is inessential if cuts a disk in . Say is peripheral (aka boundary parallel) if cuts an annulus in . Say is isotopic to , denoted by , if there is a homeomorphism isotopic to identity which maps to .

Definition 1equivalence classes of essential non-peripheral curves in equivalence classes of essential non-peripheral arcs and curves in .Add an edge in any time vertices can be represented by disjoint curves. This produces a graph called the complex of curves in .

Define distance minimal number of edges in an edge path from to in .

It is locally infinite. The link of a vertex is roughly speaking .

Theorem 2 (Masur, Minsky)is Gromov hyperbolic.

Note is quasiisometric to .

**Goal**: Assemble hyperbolicity of , a subsurface of , into information about .

**3. Train tracks **

Definition 3 (Williams, Thurston)Say a -smooth graph is a train-track is

- is locally modelled on train track switches.
- All components of have negative Euler characteristics (i.e. no no-gons, mono-gons, bi-gons, punctured disks and annuli).
- is birecurrent and large.

**The point**: A long simple closed geodesic (in a hyperbolic metric) looks like a train track.

So this helps to understand images of simple closed curves by large elements of .

Definition 4Say is carried by (denoted by ) if there is a homotopy smoothly pushing onto .

Note it can’t be an isotopy.

Definition 5Say is dual to (denoted by ) if is transverse (in particular misses the switches) and there are no bi-gons between and .

Definition 6Let denote the set of curves carried by and the set of curves dualto .

In , and are somewhat transverse. There boundaries are remote from each other, except from a small region where they intersect.

**Point**: Define to be the short curves (so a finite set). Similarly, one could define . They turn out to be coarsely identical.

** 3.1. Splitting and folding **

Definition 7Splitting a train track duplicates a branch between two inward switches. There are three ways of doing it,

- central, inserting nothing;
- left, inserting a connection from right to left branch;
- right, inserting a connection from left to right branch;

Folding is the reverse operation.

Note: if splits to ,

**4. Induced tracks **

Say is an essential subsurface if is essential.

Definition 8be the cover of corresponding to .

Definition 9 (Induction)Given an object (curve, train track, quadratic differential, pants decomposition…) , let and trim off all inessential or peripheral part to get .

There are technical variants of the definition (inspired by Bestvina-Feighn), depending what exactly is discarded. For train tracks, we want to discard everything that does not contain a closed curve, so we keep only the recurrent part.

Definition 10For a train track, we denote by the union of for a short curve. Define distance

** 4.1. Structure theorem **

Theorem 11For every surface , there is a constant such that for all splitting sequences , , for all essential subsurfaces , there is an interval such that

- If , then .
- For ,
- is short in ,
- is birecurrent and large,
- is a splitting/sliding/subhack of .

In other words, outside , the projection to is frozen. Inside , we may go down into inductively.

One can use this theorem to give a new proof of the following result.

Corollary 12 (Hamenst\” adt, MMS)Train track splitting sequences give quasigeodesics in which is thus quasiisometric to the space of train tracks of .

**5. Right/left endpoints **

The endpoints of occur

- Right: the last time when has diameter ;
- Left: the first time when has diameter .