A Diameter Bound For

By Margulis’ result, in our setting diameter and injectivty radius are inversely related. Thus, our theorem can also be viewed as a lower bound on injectivity radius; that is, with the above hypothesis, inj(M) > 1 R(l(P )) . It is known that that infinitely many closed, hyperbolic 3-manifolds of volume less than a given upper bound may be obtained by hyperbolic Dehn surgery on a finite list of compact manifolds. But only finitely many of these closed manifolds have diameter less than a given upper bound. Thus, our results provide a sharper version of Theorem 1.0. An outline of the proof is the following: we construct a straight 2-complex from a fundamental group presentation which maps into the manifold π1-isomorphically. Assuming the diameter of the mainifold is very large compared to the presentation length, Margulis provides us with a deep solid torus surrounding a short geodesic core. It turns out that the 2-complex cannot be homotoped to be disjoint from the solid torus. We consider the subcomplex that maps into the solid torus. From this subcomplex, we construct another 2-complex which simultaneously has a large torsion subgroup in first homology and a small triangulation. This is a contradiction.