Let $F(S)$ be the free group generated by the set $S$.
Suppose $H \subseteq F(S)$ be such that $H^{F(S)} \cap S = \emptyset$. That is, for any generator $s \in S$ we have that $s \notin gHg^{-1}$ for all $g \in F(S)$.
Equivalently, $s gH \neq g H$ for all $s \in S$ and $g \in F_{n}$.
I'd like to start investigating such groups so was wondering if there is a name/good reference for them? Are there any interesting features of such groups?
What about when $S$ is finite and $[F(S):H]$ is finite?