You establish a bound on the norm of the commuting not its rot.

Do I miss something obvious?

*[Fixed some typos I spotted in the proof of that lemma. -T]*

Lemma. For any we have .

Proof:

.

Now notice that one of the operators has rot at most for any . So, there is so that the -fold commutator has rot at most . Therefore, the word has rot at most and it is obviously non-trivial.

]]>Is it then a necessary condition that there is some sequence of k_i’s such that B^{k_i} converges to the identity for all B in SO(3)?

]]>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.

Theorem 5.2. discusses the best asymptotics that we could find. See https://arxiv.org/abs/1311.0138 for the preprint.

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