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 groupsof 3-manifolds.The key idea is to study quantitative geometric properties of the boundaries ofrelatively hyperbolic groups, such as linear connectedness.In particular, we prove a new existence result for quasi-arcs that avoid obstacles.
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