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COMBINATORIAL CONDITIONS THAT IMPLY WORD-HYPERBOLICITY FOR 3-MANIFOLDS
 

Summary: COMBINATORIAL CONDITIONS THAT IMPLY
WORD-HYPERBOLICITY FOR 3-MANIFOLDS
MURRAY ELDER1
, JON MCCAMMOND1
, AND JOHN MEIER
Abstract. Thurston conjectured that a closed triangulated 3-manifold
in which every edge has degree 5 or 6, and no two edges of degree 5 lie in
a common 2-cell, has word-hyperbolic fundamental group. We establish
Thurston's conjecture by proving that such a manifold admits a piece-
wise Euclidean metric of non-positive 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 3-manifolds
can be assigned a metric of non-positive 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 non-trivial re-
sults about 3-manifolds. The class of triangulations we consider are defined

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics