Solving equations in SO(3) : an open problem by Jan Mycielski I am interested in a problem closely related to a problem stated by Jan Mycielski in his paper Can One Solve Equations in Group?  (The American Mathematical Monthly, 1977, http://www.jstor.org/stable/2321255).
Let $p_1, q_1,...., p_m, q_m$ be fixed integers (both positive or negative).  I am interested in the image of the mapping $SO(3)\times SO(3) \to SO(3)$ given by $(X, Y)\mapsto X^{p_1} Y^{q_1} X^{p_2} Y^{q_2}\cdots X^{p_m}Y^{q_m}$.  By conjugacy and continuity, it is easy to see that the image is the set of all  rotations by angles in $[0,\alpha]$ for some $\alpha$ that depends on $p_1, q_1,...., p_m, q_m$.  Since $SO(3)$ contains a copy of the free group on two generators, $\alpha$ is always strictly positive.
Mycielski asked whether $\alpha$ is always equal to $\pi$.  I am intersted to know whether $\alpha$ is always at least $\pi/2$.
Is there anything new to be said about this problem, or is it still wide open?  I checked the papers citing Mycielski's paper, but none of them seem to have a solution.
 A: I'll post this as an answer so the question can be marked appropriately.
Corollary 3.3 in the paper Vladimir linked in the comments (arxiv.org/abs/1003.4093) says that $\alpha$ can be arbitrarily small.
A: Let me collect a number of known results: 
i) $\alpha$ can be arbitrarily small, see my paper
Andreas Thom, Convergent sequences in discrete groups, Canad. Math. Bull. 56 (2013), no. 2, 424–433. 
ii) There is some interest in estimating how small $\alpha$ can be in terms of the word length, this has been studied in the paper above, but also in
Abdelrhman Elkasapy and Andreas Thom, On the length of the shortest non-trivial element in the derived and the lower central series. J. Group Theory 18 (2015), no. 5, 793–804.
iii) In many cases $\alpha= \pi$. This has been studied in
Abdelrhman Elkasapy and Andreas Thom, About Gotô's method showing surjectivity of word maps, Indiana Univ. Math. J. 63 (2014), no. 5, 1553–1565.
and a more recent preprint
Anton Klyachko and Andreas Thom, New topological methods to solve equations over groups, arXiv:1509.01376
iv) The shortest word that I know for which $\alpha< \pi/2$ is $w=[[[XY X,YXY],[XYX,Y]],[[XYYX,YXXY],[X,Y]]]$
