Let $G$ be a finitely generated group and let $H$ be a subgroup of $G$. $H$ is a codimension-1 subgroup of $G$ if $C_{G}/H$ has more than one end, where $C_{G}$ is the Cayley graph of $G$.
Let $G$ and $H$ be finitely presented groups such that $H$ is a codimension-1 subgroup of $G$. Then do there exist manifolds of finite dimension $M_{G}$ and $N_{H}$ whose fundamental groups are $G$ and $H$ respectively such that $N_{H}$ is a codimension-1 submanifold of $M_{G}$?
