Let $Q$ be a recursively presented group. Is it possible to embed $Q$ into a finitely presented group $G$ such that the image of $Q$ is malnormal in $G$?
Note that a subgroup $H$ of $G$ is malnormal if $H^g\cap H\neq 1$ implies that $g\in H$. That is, $H$ intersects each of its proper conjugates trivially.
This seems far to strong (far too good!) to be true. However, I cannot find anything in the literature that even hints at the existence or non-existence of such an embedding.
(More generally, I am wondering if there is some kinds of embedding which preserves the centraliser of a given element of $Q$. That is, if we fix $x\in Q$ then can $Q$ be embedded in a finitely presented group $G$ such that $C_Q(x)=C_G(x)$ (or even $C_Q(x)\cong C_G(x)$)? Malnormality guarantees this, and is a more interesting question, but I thought I should mention this too...)