Engulfed subgroups of discrete groups

A subgroup $H$ of a group $G$ is said to be engulfed if there is a
finite-index subgroup $K$ other than $G$ itself such that $H<K$, or
equivalently if $H$ is not dense in the profinite topology on $G$.  In
this talk I will present a variety of methods for showing that a
subgroup of a discrete group is engulfed, and demonstrate how these
methods can be used to study finite-sheeted covering spaces of
topological spaces.