On ideal points of deformation curves of hyperbolic 3-manifolds with one cusp

Abstract

BY a hyperbolic 3-manifold we will mean an oriented complete hyperbolic 3-manifold of finite volume. In [I], Culler and Shalen investigated varieties of characters of representations of fundamental groups of hyperbolic 3-manifolds into S&(C). They showed that there corresponds an incompressible and boundary incompressible surface in the hyperbolic 3-manifold to each ideal point of an algebraic curve in its character variety. They used valuation theory and Bass-Serre’s theory of group actions on trees. The purpose of this paper is to give an explicit understanding of the restriction of the above gcncral theory to the most interesting special case of hyperbolic knot complements. Instead of character varieties, we USC deformation curves dcfincd by Thurston’s idcal triangulations and WC adopt the combinatorics developed by Nrumann and Zagicr in [3]. Our construction of incompressible surfaces is simpler and more explicit. Let N bc a hyperbolic 3-manifold with one cusp. Topologically N is homeomorphic to the interior of a compact 3-manifold with boundary homeomorphic to the 2-torus. We assume that N is decomposed into a finite union of ideal tetrahedra with mutually disjoint interiors. Such a decomposition is called an ideal triangulation of N. Associated to an idcal triangulation of N. there is an alTine algebraic curve C called a deformation curve ($2). In a neighborhood of the point corresponding to the complete hyperbolic structure, the points of C represent (incomplete) smooth hyperbolic structures on N. In 93, we dcfinc ideal points of C. We see that there corresponds a slope (an element of H,(?N) mod scalar multiplication) to each ideal point of C. We call it the boundary slope of the ideal point. In $4, for each idcal points of C, we construct a properly embedded surface in N whose boundary slope is that of the ideal point by patching together some copies of Thurston’s twisted square ([2]) contained in the ideal tetrahedra. In $5, we show that the splitting of N by the above surface is non-trivial and we can obtain an incompressible and boundary-incompressible surface in N after performing compressions on it.