Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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 $i$)?

share|improve this question

1 Answer 1

up vote 4 down vote accepted

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.

share|improve this answer
    
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. –  Anvita Mar 14 '11 at 23:27

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.