Cohomological dimension and engulfing

Given a group $G$ and a subgroup $H<G$, $H$ is said to be
engulfed in $G$ if there exists a homomorphism $\phi$ from $G$ to a
finite group such that $\phi(H)$ is properly contained in $\phi(G)$.
The statement that $H$ is engulfed in $G$ can therefore be thought of
as saying that $H$ looks smaller than $G$ from the perspective of
maps to finite groups. In this talk we will discuss the following
question: if $H$ looks smaller than $G$ from the perspective of
cohomological dimension, is $H$ necessarily engulfed? While there are
many well-known examples that show that the answer to this question is
no in general, we will show that engulfing is guaranteed when the
cohomology rings of objects called mapping solenoids associated to the
pair $H<G$ satisfy certain finiteness conditions.