## Knot groups and discreteness conditions

**Speaker:** Andrei Vesnin (Novosibirsk)**Abstract:** There are various discreteness conditions for subgroups of PSL(2,C) acting on a hyperbolic 3-space. Most of conditions are related either to algebraic structure of a group or to its geometric action. It was shown by T. Jorgensen that a subgroup of PSL(2,C) is discrete if and only if any its 2-generated subgroup is discrete. Some of necessary conditions for 2-generated groups, obtained by T. Jorgensen, F.

Gehring, G. Martin, and D. Tan, look as inequalities on traces of one generator and of a commutator of generators. We will say that a group is extreme if it gives the equality in such an inequality.

We will discuss hyperbolic knot groups and hyperbolic orbifold groups with are extreme groups for discreteness conditions (for example, the figure-eight knot and related orbifold groups). Also, we will discuss invariants of hyperbolic knots and links arising in this context.

**Bio: **Prof. Andrei Vesnin is head of the Laboratory of Applied Analysis, Sobolev Institute of Mathematics, Siberian Branch of Russian Academy of Sciences 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”. He was a visiting professor in Seoul National University in 2002 – 2004. Prof. Vesnin's reseach interests include low-dimensional topology, knot theory, hyperbolic geometry, combinatorial group theory, graph theory and applications.

He is the editor-in-chief of Siberian Electronic Mathematics Reports and a member of editorial boards of Siberian Mathematical Journal and Scientiae Mathematicae Japonicae. In 2008 Prof. Vesnin was elected to corresponding member of the Russian Academy of Sciences.