Quasihyperbolic 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 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.
 Publication:

arXiv eprints
 Pub Date:
 November 2011
 arXiv:
 arXiv:1111.2499
 Bibcode:
 2011arXiv1111.2499M
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Geometric Topology;
 Mathematics  Metric Geometry;
 20F65;
 20F67;
 51F99
 EPrint:
 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