Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 45, 2020, 139-174

QUASI-HYPERBOLIC PLANES IN RELATIVELY HYPERBOLIC GROUPS

John M. Mackay and Alessandro Sisto

University of Bristol, School of Mathematics
Bristol, United Kingdom; john.mackay 'at' bristol.ac.uk

ETH Zurich, Department of Mathematics
8092 Zurich, Switzerland; sisto 'at' math.ethz.ch

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.

2010 Mathematics Subject Classification: Primary 20F65, 20F67, 51F99.

Key words: Relatively hyperbolic group, quasi-isometric embedding, hyperbolic plane, quasi-arcs.

Reference to this article: J. M. Mackay and A. Sisto: Quasi-hyperbolic planes in relatively hyperbolic groups. Ann. Acad. Sci. Fenn. Math. 45 (2020), 139-174.

Full document as PDF file

https://doi.org/10.5186/aasfm.2020.4511