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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.

YCor
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Alexander Belov
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  • Why doesn't $X$=identity, $Y$=rotation by $alpha/(q_1+q_2+\ldots+q_m)$ give any rotation we desire? – Watson Ladd Jan 27 '16 at 06:01
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    @Watson, the interesting case is when the p's sum to zero and the q's sum to zero. – Peter McNamara Jan 27 '16 at 06:42
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    If I remember correctly, Borel proved for any connected semisimple algebraic group that the image of this map is Zariski dense. Would not that settle your question, more or less? – Vladimir Dotsenko Jan 27 '16 at 08:45
  • Related: http://mathoverflow.net/questions/45483/word-maps-on-compact-lie-groups. The key word is "word map". – YCor Jan 27 '16 at 10:04
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    @VladimirDotsenko Borel's result does not help, since for every $\alpha>0$ the set of rotations of angle in $[0,\alpha]$ is Zariski dense. – YCor Jan 27 '16 at 10:11
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    @YCor : yes you're right of course. In the MO question you link, there is a paper of Andreas Thom mentioned in one of the comments (http://arxiv.org/abs/1003.4093), discussion of Remark 3.6 in that paper is relevant for the case of SO(3), and shows that probably there are more conjectures than actual results for this question. – Vladimir Dotsenko Jan 27 '16 at 10:58
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    If $X$ is a rotation with angle $\beta\in (0,\pi/2)$ then $X^{\lceil \pi/(2\beta) \rceil}$ is a rotation with angle at least $\pi/2$ so $\alpha$ is at least $\pi/2$. – user35593 Jan 27 '16 at 13:07
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    Corollary 3.3 in the paper Vladimir linked (arxiv.org/abs/1003.4093) appears to answer the question: $\alpha$ can be arbitrarily small. – Ash Malyshev Jan 27 '16 at 20:20
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    @AntonMalyshev : It seems you are right! Thank you a lot, and also YCor and VladimirDotsenko! If you post your comment as an answer, I will accept it. – Alexander Belov Jan 28 '16 at 09:48
  • Thanks for posting. I was not aware that this was asked by Mycielski. – Andreas Thom Jan 31 '16 at 14:23
  • @VladimirDotsenko: Why do you think there are more conjectures than actual results? – Andreas Thom Jan 31 '16 at 14:24
  • I would replace "by conjugacy and continuity" to "by conjugacy and continuity and compactness", to resolve the possible answer $[0,\alpha)$. And a naive question: are there examples of similar questions on non-compact groups with not closed answer, like 'rotations by angle different from $\pi$'? – Fedor Petrov Jan 31 '16 at 15:18
  • @AndreasThom - I seem to have misread it, sorry. I am glad though that me mentioning your paper somehow it brought this thread to an answer :) – Vladimir Dotsenko Jan 31 '16 at 19:26

2 Answers2

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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 $w=[[[XY X,YXY],[XYX,Y]],[[XYYX,YXXY],[X,Y]]]$

Andreas Thom
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  • (From the other post): in i): see esp. Corollary 3.3. In iv) convention is $[x,y]=xyx^{-1}y^{-1}$. – YCor Jan 31 '16 at 15:06
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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.

Ash Malyshev
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