Abstract:

The use of volume of simplices in hyperbolic geometry for combinatorial problems, initiated by Thurston, has proven successful for minimal triangulations of polytopes. For example, the best lower bound so far for the minimum number of n-dimensional simplices in a triangulation of the n-cube was found by Smith as the ratio of the hyperbolic volume of the ideal cube to the ideal regular simplex in hyperbolic n-space. I will show how to use volumes in products of hyperbolic planes to give lower bounds for the minimal number of simplices needed in a triangulation of the product of two convex polygons, improving lower bounds by Bowen, De Loera, Develin and Santos.