# Commutator subgroup of a surface group

Let $\Sigma_{g,n}$ denote a compact orientable genus $g$ surface with $n$ boundary components. Assume that $g \geq 1$ and fix a basepoint $p \in \Sigma_{g,n}$. Define $S \subset [\pi_1(\Sigma_{g,n},p),\pi_1(\Sigma_{g,n},p)]$ to be the set of all elements of $\pi_1(\Sigma_{g,n},p)$ that can be realized by simple closed $p$-based curves $\gamma$ that separate $\Sigma_{g,n}$ into two components, one of which is homeomorphic to a one-holed torus with boundary component $\gamma$. It is not hard to show that if $p$ is in the interior of $\Sigma_{g,n}$, then $S$ generates $[\pi_1(\Sigma_{g,n},p),\pi_1(\Sigma_{g,n},p)]$. For instance, this is Lemma A.1 in my paper "Cutting and Pasting in the Torelli Group", though I'm sure that many other people have observed this over the years.

Assume now that $p$ lies on one of the boundary components of $\Sigma_{g,n}$. Question : Does $S$ still generate $[\pi_1(\Sigma_{g,n},p),\pi_1(\Sigma_{g,n},p)]$?

The thing that makes this difficult is the following. If $\gamma \in \pi_1(\Sigma_{g,n},p)$ can be realized by a simple closed curve and $p$ is in the interior of the surface, then all conjugates of $\gamma$ can also be realized by simple closed curves. However, this is false if $p$ is on the boundary.

I suspect that the answer is no, so here is a more general question. Continue to assume that $p$ lies on one of the boundary components of $\Sigma_{g,n}$. The group $\text{Diff}(\Sigma_{g,n},p)$ acts on $\pi_1(\Sigma_{g,n},p)$, and $S$ consists of a single orbit under this action. Question : does there exist a finite subset $T$ of $[\pi_1(\Sigma_{g,n},p),\pi_1(\Sigma_{g,n},p)]$ such that the $\text{Diff}(\Sigma_{g,n},p)$-orbit of $T$ generates $[\pi_1(\Sigma_{g,n},p),\pi_1(\Sigma_{g,n},p)]$?

Funny things might happen in the low-genus cases, so I should mention that I'm most interested in the case where $n=1$ and $g$ is large.

EDIT: I just got an up vote on this ancient unanswered question, so I thought I'd point out to anyone who is intrigued by it that my motivation for it was to find small generating sets for the Torelli group. I still don't know the answer to the question, but I was able to find a hack to avoid needing one. See my paper

Putman, Andrew, Small generating sets for the Torelli group. Geom. Topol. 16 (2012), no. 1, 111–125.

The weird arguments in Section 4 are designed to circumvent the issue identified by this question. I could probably improve the bounds in the above paper if someone gave me a reasonable answer to this question, so please let me know if you find one!

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I'm confused by the notation. Can you clarify what you mean by $[\pi_1(\Sigma_{g,n},p), \pi_1(\Sigma_{g,n},p)]$? – Bill Thurston Sep 11 '10 at 19:09
I just mean the commutator subgroup of the surface group, ie the subgroup of $\pi_1(\Sigma_{g,n},p)$ generated by all commutators $[x,y]=x^{-1} y^{-1} x y$ for $x,y \in \pi_1(\Sigma_{g,n},p)$. – Andy Putman Sep 11 '10 at 19:25