TITLE: Mechanisms generating free subgroups of Artin braid groups
QUESTION [4 upvotes]: Within my limited experience, I have only known free groups to occur through two mechanisms: as fundamental groups of trees (graphs) and ping-pong. And sometimes only through one way: the fact that sufficiently high-powers of hyperbolic elements in a Gromov-hyperbolic group generate a free group arises via ping-pong. I know of no tree to represent the situation. 
To the experts, the following question is surely either an obvious yes or no: 
Can we explicitly describe all mechanisms by which finitely generated free subgroups of the Artin braid groups $B_n$ (for $n=2,3,\ldots$) arise ? 
More specifically, seeing $B_n$ as the isotopy space of all $n$-pointed braids in the closed 2-disk, can we characterize all configurations generating the rank $k$ free group?
Much less specifically, do we know, say, that all finitely-generated free subgroups arise from ping-pong and can we describe all ping-pong games?

REPLY [1 votes]: A couple of remarks: 
If a subgroup is torsion-free, and intersects the pure braid group $P_n$ in a free group, then the group is also free. So the question can be interpreted in $P_n$. 
There is the Birman exact sequence $F_{n-1} \to P_n \to P_{n-1}$, which is obtained as deleting a puncture, and has kernel the fundamental group of an $n-1$-punctured plane. There is such a free subgroup for each puncture, and these subgroups are not ``quasiconvex" in any sense since they are normal. So I don't think there is some sort of ping-pong description of these from some action of $P_n$ on a nice space. 
On the other hand, if the image of the subgroup generated by $g_1,\ldots,g_k$ in $P_{n-1}$ is free of rank $k$, then so is the subgroup of $P_n$. So I could imagine some inductive description of the free subgroups of $P_n$. 
Suppose the image in $P_{n-1}$ is free. Then there is a Nielsen transformation of $g_1,\ldots,g_k$ such that $g_1,\ldots,g_j$ have image in $P_{n-1}$ generating a rank $j$ free subgroup, and $g_{j+1},\ldots,g_k$ have trivial image in $P_{n-1}$. Then the question is what is the extension of the free group $\langle g_1,\ldots,g_j\rangle$ by the normal subgroup of $\langle g_1,\ldots,g_n\rangle$ generated by $g_{j+1},\ldots, g_n$? I wouldn't expect a simple answer to this question though, so this is probably a difficult question. Also, the image in $P_{n-1}$ might not be free.
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