James Howie in the paper "The p-adic topology on a free group:a counterexample" showed that in the free group $F$ generated by $x$ and $y$,if $a=xy^2$, $b_1=x^{-2}y^{-3}$ and $b_2=x^{-2}(xy)^5$, then the equation $a=v_1^{-1}b_1v_{1} v_2^{-1}b_2v_{2}$ has not solution in the free group but it has solution in the free pro-p group where $v_1$ and $v_2$ are variables. I need to know what is the solution in the free pro-p group but I do not know how. I appreciate any idea.
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$\begingroup$ At the risk of asking the obvious, have you tried looking in Howie's paper? $\endgroup$– HJRWJun 15, 2015 at 12:57
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$\begingroup$ Yes I do, but he construct a Cauchy sequence by induction. I am trying to follow the induction and construct this Cauchy sequence but it was very complicated $\endgroup$– user182085Jun 15, 2015 at 13:03
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