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 .
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 1 equivalence 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 4 Say is carried by (denoted by ) if there is a homotopy smoothly pushing onto .
Note it can’t be an isotopy.
Definition 5 Say is dual to (denoted by ) if is transverse (in particular misses the switches) and there are no bi-gons between and .
Definition 6 Let 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 7 Splitting 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 8 be 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 10 For a train track, we denote by the union of for a short curve. Define distance
4.1. Structure theorem
Theorem 11 For 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 .