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Let $F$ be a free group of finite rank $r>1$ and let $c_1,c_2,...$ be a full set of basic commutators for $F$. Upon completing the $k$-th step of the collection process for an arbitrary $g\in F$, one obtains a representation

$g=c_1^{n_1}c_2^{n_2}\ldots c_{t}^{n_t}g_k$,

where

$c_1,\ldots,c_t$ are all basic commutators of weight less than $k$ ($t$ depending on $k$),

$n_1,\ldots,n_t$ are integers,

$g_k\in \gamma_k$, the $k$-th term of the lower central series for $F$.

Question. Is there always $g\in G$ such that the series $g_1,g_2,\ldots$ never terminates (i.e. has nonidentity terms $g_i$ for arbitrarily large $i$)?

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Yes. I believe (ab)^3 or so, at least (ab)^n for high enough n, never terminates. As far as I recall, only (ab)^1 and (ab)^2 terminate. If you start to collect them, I think you'll see the proof.

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  • $\begingroup$ I cannot say that I see the proof, but the rate of growth of the expression when collecting $(ab)^3$ even after step $1$ shatters any hope of it ever terminating. It is enough for me to know that the answer is affirmative, though. Thanks. $\endgroup$
    – Anvita
    Commented Mar 14, 2011 at 23:27

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