Closed incompressible surfaces in 2-generator hyperbolic 3-manifolds with a single cusp

Abstract

A knot K is said to have tunnel number 1 if there is an embedded arc A in S3 with endpoints on K, whose interior is disjoint from K and such that the complement of a regular neighbourhood of K u A is a genus 2 handlebody. In particular the fundamental group of the complement of a tunnel number one knot is 2-generator. There has been some interest in the question as to whether there exists a hyperbolic tunnel number one knot whose complement contains a closed essential surface. The aim of this paper is to prove the existence of infinitely many 2-generator hyperbolic 3-manifolds with a single cusp which contain a closed essential surface. One such example is a knot complement in RP3. The methods used are of interest as they include the possibility that one of our examples is a knot complement in S3. © 1993, Edinburgh Mathematical Society. All rights reserved.