Title：Volumes of generalized hyperbolic polyhedra and hyperbolic links
Reporter：Andrei Vesnin
Work Unit：Sobolev Institute of Mathematics
Time：2023/08/20 9:30-10:30
Address：Room 313, Zhengxin Building
Summary of the report: A polyhedron in a three-dimensional hyperbolic space is said to be generalized if finite, ideal and truncated vertices are admitted. In virtue of Belletti's theorem (2021) the exact upper bound for volumes of generalized hyperbolic polyhedra with the same one-dimensional skeleton G is equal to the volume of an ideal right-angled hyperbolic polyhedron whose one-dimensional skeleton is the medial graph for G. We will present the upper bounds for the volume of an arbitrary generalized hyperbolic polyhedron, where the bonds linearly depend on the number of edges. Moreover, it is shown that the bounds can be improved if the polyhedron has triangular faces and trivalent vertices. As an application there are obtained new upper bounds for the volume of the complement to the hyperbolic link having more than eight twists in a diagram. The results under discussion are based on the preprint arXiv:2307.04543 (https://arxiv.org/abs/2307.04543).
Introduction of the Reporter:
Professor Andrei Vesnin is head of the Laboratory of Applied Analysis, Sobolev Institute of Mathematics and a professor of Geometry and Topology, Novosibirsk State University. He received a Candidate of Sciences in physics and mathematics is 1991 from Sobolev Institute of Mathematics for the thesis “Discrete groups of reflections and three-dimensional manifolds”, and a Doctor of Sciences in physics in mathematics in 2005 for the thesis “Volumes and isometries of three-dimensional hyperbolic manifolds and orbifolds”.
Professor Vesnin's research interests include low-dimensional topology, knot theory, hyperbolic geometry, combinatorial group theory, graph theory and applications. In 2008, Prof. Vesnin was elected to corresponding member of the Russian Academy of Sciences. He is a member of the editorial board of the journal "International Journal of Mathematics and Computer Science", doctoral dissertation council at the Institute of Mathematics SB RAS.
