# Cusp size bounds from singular surfaces in hyperbolic 3-manifolds Joint work with Colin Adams, Adam Colestock, William Gillam, Eric Katerman

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9.

We also find particular upper bounds on meridian length, $\ell$-curve length and maximal cusp volume for hyperbolic knots in $S^3$ depending on crossing number. Particular improved bounds are obtained for alternating knots.

### Bibliographic details

Adams, C. and Colestock, A. and Fowler, J. and Gillam, W. and Katerman, E. (2006). Cusp size bounds from singular surfaces in hyperbolic 3-manifolds. Trans. Amer. Math. Soc., Volume 358, 727–741 (electronic).