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Is there a group $G$ with a subgroup $H\subseteq G$ of finite index and an element $g\in G$ such that $H\cap g^{-1} H g$ has infinite index in $G$?

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This is not possible, since $[G: H \cap K] \leq [G:H] \cdot [G:K]$ if $H$ and $K$ are subgroups of $G$.

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