Seminar Algebra and Geometry: Matthieu Jacquemet (Université de Fribourg)

Hyperbolic volume, estimates, and simplices

We are interested in estimating the volume of a hyperbolic n-polyhedron by knowing its f-vector, and its 'inradius'. In fact, exact volume formulas are difficult to find for n>2. However, in dimension 3, there is a volume estimate by Fejes-Toth (for spaces of constant curvature) in terms of the inradius and the f-vector, showing some deficiencies. A natural question in this context (but also in general) is the inradius of a hyperbolic simplex. I shall start with a (partial) overview of the context of hyperbolic orbifolds/manifolds and their volume. In a second part, I shall discuss Fejes-Toth's result and give some new results for compact simplices in any dimensions.

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