The bumping set and the characteristic submanifold

TL;DR

This paper establishes a connection between the Nielsen core of the bumping set in Kleinian groups and the boundary of the characteristic submanifold in 3-manifolds, providing new insights and a direct construction method.

Contribution

It introduces a novel relationship between the bumping set's Nielsen core and the characteristic submanifold, with a new construction approach from first principles.

Findings

Nielsen core of bumping set equals boundary of characteristic submanifold

Provides examples of characteristic submanifolds

Constructs characteristic submanifold directly from Nielsen core

Abstract

We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group $\Gamma$ is the boundary of the characteristic submanifold of the associated 3-manifold with boundary. Some examples of interesting characteristic submanifolds are given. We also give a construction of the characteristic submanifold directly from the Nielsen core of the bumping set. The proofs are from "first principles", using properties of uniform domains and the fact that quasi-conformal discs are uniform domains.