Parabolic cut pairs in boundaries of relatively hyperbolic groups

TL;DR

This paper introduces a new phenomenon called parabolic cut pairs in the boundaries of relatively hyperbolic groups, providing a construction method and characterizing all such groups through a combination theorem.

Contribution

It develops a novel combination theorem for relatively hyperbolic groups with inseparable parabolic cut pairs and offers explicit topological descriptions of their boundaries.

Findings

Constructed examples of relatively hyperbolic groups with parabolic cut pairs.

Proved that all such groups arise via the proposed combination theorem.

Provided explicit topological descriptions of the boundaries.

Abstract

Parabolic cut pairs in the boundaries of relatively hyperbolic group are a new and previously unexplored phenomenon. In this paper, we give a way to create examples of relatively hyperbolic groups with parabolic cut pairs on their boundary via a combination theorem, which states that a group $G$, splitting as a graph of relatively hyperbolic groups with certain conditions, is relatively hyperbolic with inseparable parabolic cut pairs on the boundary $\partial(G,\mathcal{P})$. We also prove that all relatively hyperbolic groups with inseparable parabolic cut pairs in their boundaries arise via this combination theorem. \'Swi\k{a}tkowski gives a topological description of combining boundaries of vertex groups. Unfortunately, his method cannot be applied for fundamental reasons in this setting. We instead give two explicit topological descriptions of the boundary in terms of boundaries of the vertex groups.