## Notes of Saul Schleimer’s talk

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 ${Out(F_n)}$.

1. Introduction

Consider surface ${S}$ of genus ${g}$ and ${n}$ boundary components.

There are several spaces on which mapping class group ${\mathcal{MCG}(S)}$ of ${S}$ act. We want to understand the coarse geometry of these spaces.

2. Arcs and curves

A curve ${\alpha}$ is a proper embedding of ${S^1}$ into ${S}$. An arc ${\alpha}$ is a proper embedding of interval ${I}$ into ${S}$. Say ${\alpha}$ is inessential if ${\alpha}$ cuts a disk in ${S}$. Say ${\alpha}$ is peripheral (aka boundary parallel) if ${\alpha}$ cuts an annulus in ${S}$. Say ${\alpha}$ is isotopic to ${\beta}$, denoted by ${\alpha\sim\beta}$, if there is a homeomorphism ${f}$ isotopic to identity which maps ${\alpha}$ to ${\beta}$.

Definition 1 ${C^0 (S)=}$ equivalence classes of essential non-peripheral curves in ${S}$ ${AC^0 (S)=}$ equivalence classes of essential non-peripheral arcs and curves in ${S}$.

Add an edge in ${C^1 (S)}$ any time vertices can be represented by disjoint curves. This produces a graph ${C(S)}$ called the complex of curves in ${S}$.

Define distance ${d_S (\alpha,\beta)=}$ minimal number of edges in an edge path from ${\alpha}$ to ${\beta}$ in ${C(S)}$.

It is locally infinite. The link of a vertex ${\alpha}$ is roughly speaking ${C(S\setminus \alpha)}$.

Theorem 2 (Masur, Minsky) ${C(S)}$ is Gromov hyperbolic.

Note ${AC(S)}$ is quasiisometric to ${C(S)}$.

Goal: Assemble hyperbolicity of ${C(X)}$, ${X}$ a subsurface of ${S}$, into information about ${\mathcal{MCG}(S)}$.

3. Train tracks

Definition 3 (Williams, Thurston) Say a ${C^1}$-smooth graph ${\tau\subset S}$ is a train-track is

1. ${\tau}$ is locally modelled on train track switches.
2. All components of ${S\setminus\tau}$ have negative Euler characteristics (i.e. no no-gons, mono-gons, bi-gons, punctured disks and annuli).
3. ${\tau}$ 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 ${\mathcal{MCG}(S)}$.

Definition 4 Say ${\alpha}$ is carried by ${\tau}$ (denoted by ${\alpha < \tau}$) if there is a homotopy smoothly pushing ${\alpha}$ onto ${\tau}$.

Note it can’t be an isotopy.

Definition 5 Say ${\alpha}$ is dual to ${\tau}$ (denoted by ${\alpha \not\cap \tau}$) if ${\alpha}$ is transverse (in particular misses the switches) and there are no bi-gons between ${\alpha}$ and ${\tau}$.

Definition 6 Let ${C(\tau)\subset C(S)}$ denote the set of curves carried by ${\tau}$ and ${C^*(\tau)\subset C(S)}$ the set of curves dualto ${\tau}$.

In ${C(S)}$, ${C(\tau)}$ and ${C^* (\tau)}$ are somewhat transverse. There boundaries are remote from each other, except from a small region where they intersect.

Point: Define ${V(\tau)\subset C(\tau)}$ to be the short curves (so a finite set). Similarly, one could define ${V^* (\tau)}$. 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,

1. central, inserting nothing;
2. left, inserting a connection from right to left branch;
3. right, inserting a connection from left to right branch;

Folding is the reverse operation.

Note: if ${\tau}$ splits to ${\sigma}$,

$\displaystyle \begin{array}{rcl} C(\sigma)\subset C(\tau),\quad C^*(\tau)\subset C^* (\sigma). \end{array}$

4. Induced tracks

Say ${X\subset S}$ is an essential subsurface if ${\partial X \subset X}$ is essential.

Definition 8 ${\rho_X :S^{X}\rightarrow S}$ be the cover of ${S}$ corresponding to ${\pi_1 (X)\subset \pi_1 (S)}$.

Definition 9 (Induction) Given an object (curve, train track, quadratic differential, pants decomposition…) ${\alpha\in S}$, let ${\alpha^X =\rho_X^{-1}}$ and trim off all inessential or peripheral part to get ${\alpha_{|X}\in AC(X)}$.

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 ${\tau}$ a train track, we denote by ${\pi_X (\tau)}$ the union of ${\alpha_{|X}}$ for ${\alpha\in V(\tau)}$ a short curve. Define distance

$\displaystyle \begin{array}{rcl} d_X (\tau,\sigma)=d_X (V(\tau),V(\sigma)). \end{array}$

4.1. Structure theorem

Theorem 11 For every surface ${S}$, there is a constant ${K}$ such that for all splitting sequences ${(\tau_i)}$, ${i=0,...,N}$, for all essential subsurfaces ${X\subset S}$, there is an interval ${I_N \subset[0,N]}$ such that

1. If ${[a,b]\cap I_X =\emptyset}$, then ${d_X (\tau_a,\tau_b)\leq K}$.
2. For ${i\in I_X}$,
3. ${\partial X}$ is short in ${\tau_i}$,
4. ${{\tau_i }_{|X}}$ is birecurrent and large,
5. ${{\tau_{i+1} }_{|X}}$ is a splitting/sliding/subhack of ${{\tau_i }_{|X}}$.

In other words, outside ${I_X}$, the projection to ${X}$ is frozen. Inside ${I_X}$, we may go down into ${X}$ 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 ${\mathcal{MCG}(S)}$ which is thus quasiisometric to the space ${TT(S)}$ of train tracks of ${S}$.

5. Right/left endpoints

The endpoints of ${I_X}$ occur

• Right: the last time when ${C(\tau_{|X})}$ has diameter ${\geq 3}$;
• Left: the first time when ${C^* (\tau_{|X})}$ has diameter ${\geq 3}$.