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I have some question about the last conclusion in the proof of Lemma 2.16 on page 20 of the paper "Hall's theorem for limit groups". You can find the paper on

https://arxiv.org/abs/math/0605546

So. After assuming that for infinitely many indices $p$, $q_p$ and some $b_k$ we have that

$$z^p[b_k,z^{q_p}]z^{-p}\in\pi_1(Y')$$

the author concludes that $[b_k,z]=1$, since the elevations $\partial_{1,2}^+$ are proper. If i am right, he applies the previous Lemma, Lemma 2.15 on page 19. But to do so, i think, one has to assume that there exists some fixed $g\in G$ such that for infinitely many indices $p$ the element $g_p=[b_k,z^{q_p}]$ equals $g$. But how is that possible? The element $g_p$ depends on $p$. So couldn't it happen that for any $p,p'$ with $p\neq p'$ we have $g_p\neq g_{p'}$?

Thanks for your response!

PS: I'm sorry, if this is rather a question for mathstackexchange.

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    $\begingroup$ Did you email Henry? He's currently at Cambridge (UK). $\endgroup$
    – Kevin Buzzard
    Commented Jan 4, 2017 at 17:16
  • $\begingroup$ Sorry. I can't write comments yet. I wrote him an e-mail. $\endgroup$
    – Peter
    Commented Jan 4, 2017 at 22:06
  • $\begingroup$ You can always write comments under your own post. $\endgroup$
    – Todd Trimble
    Commented Jan 5, 2017 at 1:29
  • $\begingroup$ Okay. Thanks. May the problem was, that I tried to write some comment with my mobile phone. $\endgroup$
    – Peter
    Commented Jan 5, 2017 at 5:32

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