Quasi-hyperbolic planes in relatively hyperbolic groups

Details Quasi-hyperbolic planes in relatively hyperbolic groups Quasi-hyperbolic planes in relatively hyperbolic groups

2011-11-10

arxiv.org/abs/1111.2499v1

Mathematics

Group Theory

32 pages, 4 figures

Scientific paper

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit certain splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. The specific embeddings we find remain quasi-isometric embeddings when composed with the natural 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.

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