Multiple Dehn twists and minimal position - MathOverflow

Multiple Dehn twists and minimal position

2012-06-02T16:33:31Z

I have a question about a proof that I am reading in "A primer on Mapping Class Groups" by Farb and Margalit.

Let $a$ be a simple closed curve in a compact surface $S$ (possibly with marked points and boundary components) not isotopic to a point or boundary component and let $T_a$ denote the Dehn twist about $a$.

Now let $\alpha_1 , ..., \alpha_n$ be a collection of pairwise disjoint isotopy classes of simple closed curves in $S$ and let $M = \prod_{i=1}^{n} T_{\alpha_i}^{e_i}$. Also suppose that $e_i >0 $ $\forall i$ or $e_i <0$ $\forall i$ where $e_i$ is an integer and that $b$ is an arbitrary isotopy class of a simply closed curve.

Ok now we look at $M(b)$, and find a representative $\beta '$ in its isotopy class. We also take $\beta$ to be in the isotopy class of $b$.

I now want to show that $\beta$ and $\beta'$ are in minimal position!

For context, this is propostion 3.4 in Farb and Margalit's "A primer on Mapping Class Groups".

If one lets $n=1$ above then it wouldn't be hard to prove (prop 3.2 in the same book) as it follows from the bigon criterion. However they say that it also is true here for $\beta$ and $\beta'$ since all the $e_i$'s have the same sign or rather all the twists are in the same direction.

So what happens if we allow the $e_i$'s to have arbitrary sign?

Any comments/help would be greatly appreciated! cheers!

P.S. I have asked this at math stack exchange but thought I would ask at this site as well.

Answer by Sam Nead for Multiple Dehn twists and minimal position

2012-06-02T17:15:30Z

First (and trivially), yes, there are representatives $b$ and $b'$ of the curve and its image that are in minimal position. But you aren't really asking this. Correct me if I am wrong, but you are actually asking "Is there a small motion of the image of $b$ placing it in minimal position to the original position of $b$?" Said original position is assumed to be minimal with respect to the curves $\{a_i\}$.

The answer is "no". There is no such small motion. To see this, consider such $b$ and $\{a_i\}$ so that there are two components of $b - n(\cup a_i)$ that are parallel as proper arcs in $S - n(\cup a_i)$.

Answer by Dan Margalit for Multiple Dehn twists and minimal position

2012-07-06T13:13:09Z

Ivanov gives a formula for the case where the exponents are arbitrary integers. It is Lemma 4.2 on page 39 of his book Subgroups of Teichmuller Modular Groups. He says "natural numbers" in the statement, but he means "integers".