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> Are the pure genus zero mapping class groups residually torsion-free nilpotent?
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They are 


-a quotient of the pure braid groups (which are residually torsion-free nilpotent). 


-torsion-free.



Are the pure genus zero mapping class groups residually torsion-free nilpotent?
The answer is yes. This is because these groups are fundamental groups of complements of fiber-type hyperplane arrangements; the fact that such groups are residually torsion free nilpotent goes back to Falk and Randell. 


Indeed we are considering the fundamental group of the moduli space $M\_{0,n}$ of genus zero Riemann surfaces with $n$ distinct ordered marked points. Since the group of Möbius transformations acts 3-transitively on the Riemann sphere, we can take the first three markings to be $0,1,\infty$. This identifies $M\_{0,n}$ with the space $$\{(x\_1,\ldots,x\_{n-3}) \in \mathbf C^{n-3} : x\_i \neq x\_j, x\_i \neq 0, x\_i \neq 1 \text{ for all }i,j\},$$
which is of the required form.


