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Quasi-hyperbolic planes in relatively hyperbolic groups

Research output: Contribution to journalArticle

Original languageEnglish
JournalAnnales Academiae Scientiarum Fennicae Mathematica
DateAccepted/In press - 12 Apr 2019

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.

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