Compressed subgroup

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is termed compressed if for any subgroup $$K$$ of $$G$$ containing $$H$$, the rank of $$H$$ is not more than the rank of $$K$$.

We typically study compressed subgroups inside a free group.

In terms of the intermediately operator
The subgroup property of being compressed is obtained by applying the intermediately operator to the subgroup property of being a rank-dominated subgroup. A subgroup is said to be rank-dominated in a group if the rank of the subgroup is not more than the rank of the group.

Stronger properties

 * Inert subgroup

Weaker properties

 * Finite-index-maximal subgroup in case we are working within a free group

Transitivity
It is not clear whether any compressed subgroup of a compressed subgroup is compressed. The problem lies with the fact that intermediate subgroups containing the smaller subgroup may not be comparable either way with the bigger subgroup.

If a subgroup is compressed in terms of the whole group, it is clearly compressed in every intermediate subgroup.

Clearly, the trivial subgroup is compressed in any group, because it is generated by zero elements. Also, every group is compressed as a subgroup of itself.