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\title{Dynamics Seminar\\March 3, 2006}
\author{Gabriel C. Drummond-Cole}
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\begin{document}
\maketitle

Grafting coordinates for Teichmueller space

I'm going to talk about grafting and its relation to Teichmueller space, how it can give you coordinates.

I am going to start by talking about grafting.  Let me say once and for all that $S$ is a compact surface of genus $g>1$ and $T(S)$ is the Teichmueller space of marked hyperbolic or sometimes complex or comformal structures ($S$ is smooth).  This is a $6g-6$ dimensional cell or a $3g-3$ himensional complex manifold.  You start with some metric and you take $\gamma$ an isotopy class of simple closed curves which we identify with the symple closed geodesics.  We will cut it out and replace it with a cylinder which is the product of $\gamma$ and an interval of length $t.$  This has a well defined metric which is not fully hyperbolic.  So this is $gr_{t\gamma}:T(S)\to T(S).$  Secretly I think of $t$ as a weight assigned to a geodesic $\gamma.$  So we allow $\gamma$ to vary and we get a map $gr:\mathrm{weighted\ simple\ closed\ curves}\times T(S)\to T(S).$

Now a geodesic lamination is a foliation of a closed subset of $X$ by complete simple hyperbolic geodesics.  On a compact surface this is a strong condition.

There is a canonical way of identifying two for different metrics, it is just convienient to use the metric.

A transverse measure for a $\Lambda$ which is a geodesic lamination assigns a Borel measure to each transverse arc $k:[a,b]\to S$ compatible with splitting and isotopy of transversals.

A measured geodesic lamination $\lambda=(\Lambda,\mu)$ is a geodesic lamination and a transverse measure of full support.  This means we can't have spiraling geodesics.

This means if $k$ is a transversal then the total measure of $k$ is what we call $i(\lambda,k).$

A sequence of measured laminations converge to $\lambda$ if $i(\lambda_n,k)\to i(\lambda,k)$ for all compact $k.$

A few facts: $\mathscr{ML}$ (the set of measured geodesic laminations) $\cong \mathbb{R}^{6g-6}$ and finitely many tranversals $k_1,\ldots,k_N$ give you an embedding $\mathscr{ML}\injects\mathbb{R}^N.$

What are you going to do with these things?  Well, $t\gamma$ are dense in $\mathscr{ML}.$  So it is the completion of the set of simple closed geodesics.  You can approximate a sum of two with a single one.  

\begin{thm}
(Thurston)
\\Grafting has a (unique) continuous extension to $\mathscr{ML},$ i.e., $gr:\mathscr{ML}(S)\times T(S)\to T(S).$
\end{thm}

The main theorem
\begin{thm}(D, Wolf)\\
For any $X\in T(S)$ the map $\lambda\mapsto gr_\lambda X.$
\end{thm}

Let me just point out that $\mathscr{ML}(S)\to 0,$ the empty lamination, is the origin for the cone structure on $\mathscr{ML}.$  There is an action of $\mathbb{R}^+$ where you scale the measure.  

This makes $\mathscr{ML}$ a cone centered at this point you've chosen.

Clearly $gr_0(X)=X.$

You can have a single lamination that admits many coordinate systems.

I want to discuss some of the ideas that go into this theorem, but I want to discuss a cousin, the Scannell-Wolf theorem.
\begin{thm}
For a fixed $\lambda,$ the map $gr_\lambda$ is a real analytic diffeomorphism.
\end{thm}

These both have something in common as far as overall strategy.  

Local injectivity implies homeomorphism.  This is because grafting is proper, so tha


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