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The setup is that $F$ is a free finitely generated group, $H, H'$ are subgroups of index $2$, and $\tau:H\to H'$ is an isomorphism.

Denote by $B_r$ the ball around $1$ of radius $r$ in $F$, in the word metric.

Question:

Is it possible to find a sequence $G_1,\dots, G_n$ of normal subgroups of $F$ of the same index, with $n$ depending on $r$, such that

a) $G_i\cap B_r=\{1\}$, for $i=1,\dots,n$

b) $G_i\not\subset H$ and $G_i\not\subset H'$ for $i=1,\dots,n$

c) $\tau(G_i\cap H)=G_{i+1}\cap H'$ for $i=1,\dots,n-1$

d) when $r\to\infty$ then $n\to\infty.$

I am interested in general in the dynamics of such partial automorphisms $\tau$ and the situation above is the one I would most like to understand.

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I don't think much of anything at all is known about questions like this. The only paper I know about which studies specific examples of virtual automorphisms is the paper of my student Honglin Min "Hyperbolic graphs of surface groups". –  Lee Mosher Aug 26 '13 at 21:01
    
@LeeMosher Thank you, I will take a look at that paper. I have tried to start with $G_1$ and then construct such a sequence. It does not work for any $G_1$, but maybe one can find $G_1$ by using some probability considerations, random subgroups etc. –  Rostyslav Kravchenko Aug 27 '13 at 13:15
    
Why this does not work for any $G_1$? –  Kate Juschenko Aug 28 '13 at 14:26

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