Tuesday, March 27, 2018
03/27/2018 - 4:15pm to 5:15pm
The class of acylindrically hyperbolic groups consists of groups that admit a particular nice type of non-elementary action on a hyperbolic space, called an acylindrical action. This class contains many interesting groups such as non-exceptional mapping class groups, Out(Fn) forn > 1, and right-angled Artin and Coxeter groups, among many others. Such groups admit uncountably many different acylindrical actions on hyperbolic spaces, and one can ask how these actions relate to each other. Inthis talk, I will describe how to put a partial order on the set of acylindrical actions of a given group on hyperbolic spaces, which roughly corresponds to how much information about the group different actions provide. This partial order organizes these actions into a poset. I will givesome structural properties of this poset and, in particular, discuss for which (classes of) groups the poset contains a largest element.
03/27/2018 - 4:30pm to 5:30pm
Spherical Double Affine Hecke Algebra can be viewed as a noncommutative (q,t)-deformation of the SL(N,C) character variety of the fundamental group of a torus. This deformation inherits major topological property from its commutative counterpart, namely Mapping Class Group of a torus SL(2,Z) acts by atomorphisms of DAHA. In my talk I will define a genus two analogue of A1 spherical DAHA and show thatthe Mapping Class Group of a closed genus two surface acts by automorphisms of such algebra. I will then show that for special values of parameters q,t satisfying qnt2=1 for some nonnegative integer n this algebra admits finite dimensional representations. I will conclude with discussion of potential applications to TQFT and knot theory.Based on arXiv:1704.02947 joint with Sh. Shakirov