Universal surgery bounds on hyperbolic 3-manifolds. (Q1430388)

The injectivity radius of a closed hyperbolic 3-manifold \(M\) is the largest radius of an open hyperbolic ball that can be embedded at any point in \(M\). The author shows that the injectivity radius has global implications for the topology of \(M\), in terms of surgery descriptions. More precisely, the main result proved in the paper is the following. If the injectivity radius of \(M\) is at least \(2\pi\sqrt 3/5+\log (2+\sqrt 3)=3.493\ldots\), then \(M\) cannot be obtained by \(p/q\) surgery on a knot in \(S^3\) with \(| q|>4\). Moreover, no \(p/q\) surgery on a knot of genus \(g\) in \(S^3\) can give \(M\) if \(| p|\geq 5(2g-1)\).