  Given a finitely generated subgroup $H$ of a finitely generated group $G$ and
a non-principal ultrafilter $\omega$, we consider a natural subspace,
$Cone^{\omega}_{G}(H)$, of the asymptotic cone of $G$ corresponding to $H$.
Informally, this subspace consists of the points of the asymptotic cone of $G$
represented by elements of the ultrapower $H^{\omega}$. We show that the
connectedness and convexity of $Cone^{\omega}_{G}(H)$ detect natural properties
of the embedding of $H$ in $G$. We begin by defining a generalization of the
distortion function and show that this function determines whether
$Cone^{\omega}_{G}(H)$ is connected. We then show that whether $H$ is strongly
quasi-convex in $G$ is detected by a natural convexity property of
$Cone^{\omega}_{G}(H)$ in the asymptotic cone of $G$.
