  We show that any group that is hyperbolic relative to virtually nilpotent
subgroups, and does not admit peripheral splittings, contains a
quasi-isometrically embedded copy of the hyperbolic plane. In natural
situations, the specific embeddings we find remain quasi-isometric embeddings
when composed with the inclusion map from the Cayley graph to the coned-off
graph, as well as when composed with the quotient map to "almost every"
peripheral (Dehn) filling.
  We apply our theorem to study the same question for fundamental groups of
3-manifolds.
  The key idea is to study quantitative geometric properties of the boundaries
of relatively hyperbolic groups, such as linear connectedness. In particular,
we prove a new existence result for quasi-arcs that avoid obstacles.
