Quasi-hyperbolic planes in relatively hyperbolic groups

Abstract

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.

v1: 32 pages, 4 figures. v2: 38 pages, 4 figures. v3: 44 pages, 4 figures. An application (Theorem 1.2) is weakened as there was an error in its proof in section 7, all other changes minor, improved exposition

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Cited by in corpus (7)

• Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces
• On metric relative hyperbolicity
• Embedding relatively hyperbolic groups in products of trees
• Conformal dimension of hyperbolic groups that split over elementary subgroups
• Maps between relatively hyperbolic spaces and between their boundaries
• Quasi-circles through prescribed points
• On the Čech cohomology of Morse boundaries