 
Summary: COMBINATORIAL CONDITIONS THAT IMPLY
WORDHYPERBOLICITY FOR 3MANIFOLDS
MURRAY ELDER1
, JON MCCAMMOND1
, AND JOHN MEIER
Abstract. Thurston conjectured that a closed triangulated 3manifold
in which every edge has degree 5 or 6, and no two edges of degree 5 lie in
a common 2cell, has wordhyperbolic fundamental group. We establish
Thurston's conjecture by proving that such a manifold admits a piece
wise Euclidean metric of nonpositive curvature and the universal cover
contains no isometrically embedded flat planes. The proof involves a
mixture of computer computation and techniques from small cancella
tion theory.
1. Introduction
In this article we show that a class of closed triangulated 3manifolds
can be assigned a metric of nonpositive curvature. In addition to proving
a conjecture of Thurston, our main result illustrates the way in which the
computer program developed by the first and second authors ([6]) can be
used in conjunction with combinatorial methods to establish nontrivial re
sults about 3manifolds. The class of triangulations we consider are defined
