Abstract
We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasiisometrically embedded copy of the hyperbolic plane. In natural situations, the specific embeddings we find remain quasiisometric embeddings when composed with the inclusion map from the Cayley graph to the conedoff 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 3manifolds.
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 quasiarcs that avoid obstacles.
We apply our theorem to study the same question for fundamental groups of 3manifolds.
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 quasiarcs that avoid obstacles.
Original language  English 

Journal  Annales Academiae Scientiarum Fennicae Mathematica 
Publication status  Accepted/In press  12 Apr 2019 
Fingerprint Dive into the research topics of 'Quasihyperbolic planes in relatively hyperbolic groups'. Together they form a unique fingerprint.
Profiles

Dr John M Mackay
 School of Mathematics  Associate Professor in Pure Mathematics
 Probability, Analysis and Dynamics
 Pure Mathematics
 Analysis
Person: Academic , Member