Let $G$ be a finitely generated residually finite group and $\hat G$ its profinite completion. Then for all $g\in \hat G$ we have $gGg^{-1}\leq \hat G$ is dense.

Suppose that $H\leq \hat G$ is a closed subgroup such that $H\cap G$ is dense in $H$.

Can we say something in general on the density of $H\cap gGg^{-1}$ in $H$ ? Or if we assume that $G$ is a $p$-group (and so $\hat G$ a pro-$p$ group) ?

N.B. if $A\leq \hat G$ is of finite index, then $A\cap G$ is dense in $A$, hence the question is interesting only for infinite index subgroups.

  • $\begingroup$ Let $G=\mathbb{Z}*C_2$ be the free product of the infinite cyclic group with the cyclic group of order $2$. Then $H=C_2$ is a closed self-normalizing subgroup of $\widehat G$ (as it is a profinite free factor). Since $G$ has only one conjugacy class of elements of order $2$, it's easy to deduce that for any $g \in \widehat{G} \setminus G$, $g^{-1} C_2 g \cap G=\{1\}$. Therefore $C_2 \cap g G g^{-1}=\{1\}$ is not dense in $C_2$. $\endgroup$ – Ashot Minasyan May 28 '15 at 15:09

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