Quasi-hyperbolic planes in relatively hyperbolic groups

TL;DR

This paper demonstrates that certain relatively hyperbolic groups without peripheral splittings contain quasi-isometric embeddings of the hyperbolic plane, with implications for 3-manifold groups and boundary geometry.

Contribution

It establishes the existence of hyperbolic plane embeddings in a broad class of relatively hyperbolic groups and analyzes their boundary properties, extending previous results.

Findings

Existence of hyperbolic plane embeddings in specific relatively hyperbolic groups.

Quasi-isometric embeddings remain stable under natural geometric maps.

New results on boundary structures and obstacle-avoiding quasi-arcs.

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.