Paul Gustafson Thesis Defense: On the Property F Conjecture

Date: March 7, 2018

Time: 1:00PM - 2:00PM

Location: BLOC 624

Speaker: Paul Gustafson, Texas A&M University

Description: This thesis solves a question posed by Etingof, Rowell, and Witherspoon: As a modular category, $\Mod-D^\omega(G)$ gives rise to (projective) representations of mapping class groups of compact surfaces with boundary. Are the images of these representations always finite? We answer the above question in the affirmative, generalizing their work in the braid group case. Our approach is to translate the problem into manipulation of colored graphs embedded in the given surface as defined by Kirillov. To do this translation, we use the fact that any such representation associated to a finite group $G$ and 3-cocycle $\omega$ is isomorphic to a Turaev-Viro-Barrett-Westbury (TVBW) representation associated to the spherical fusion category $\text{Vec}_G^\omega$ of twisted $G$-graded vector spaces. As shown by Kirillov, the representation space for this TVBW representation is canonically isomorphic to a vector space spanned by $\text{Vec}_G^\omega$-colored graphs embedded in the surface. By analyzing the action of the Birman generators on a finite spanning set of colored graphs, we find that the mapping class group acts by permutations on a slightly larger finite spanning set. This implies that the representation has finite image.