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Let $a$, $b$, and $c$ be simple closed curves in an orientable surface $S$ such that $i(a,b) \geq 2$, $i(a,c) \geq 1$, and $i(b,c) = 0$. Let $w$ be a nontrivial element of the free group $\langle T_a,T_b \rangle$ which is different from $T_a^p$, $p$ nonzero integer, and set $x = w(a)$. If $i(a,x) > i(b,x)$ and $n$ is a nonzero integer, is there a way to compare $i(a,T_c^n(x))$ and $i(b,T_c^n(x)) = i(b,x)$ ?

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What's the motivation for this problem? – Andy Putman Apr 26 '10 at 14:09

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