Westlake Math Colloquium第四十二期 | Ben Williams: Automorphisms of Tautological Algebras on Character Varieties

2023-07-31 13:25:09



主持人:Xing Gu, Westlake University

主讲人Ben Williams, University of British Columbia

讲座主题:Automorphisms of Tautological Algebras on Character Varieties


In order to understand a finitely generated group Gamma, we can consider its representations in a linear group, such as SL(2; C). The collection of all such representations has the structure of a variety, and if we consider representations up to change-of-basis, we arrive at a quotient variety: the character variety X of the group Gamma. These have been thoroughly investigated when Gamma = pi_1(S^3 - K, m) is the fundamental group of a hyperbolic knot complement. In this case, outer automorphisms of Gamma correspond to symmetries of the knot. A difficult argument, based on Mostow rigidity, says that orientation-preserving symmetries of S^3 - K act trivially on the best-understood part of the character variety.

In order to extract information about knot symmetries from character varieties, I will consider a 'tautological' bundle of matrix algebras A on the character variety, and the induced symmetries of A arising from automorphisms of Gamma. I will construct an invariant associated to such an automorphism, depending only on the outer automorphism class, and explain how to compute it. This automorphism can be used to distinguish between outer automorphism classes. If time permits, I will explain the calculations for the figure-8 knot and its symmetries.

This represents joint work with Justin Lawrence and Nicholas Rouse.