On the Property F Conjecture

Abstract

This thesis solves the following question posed by Etingof, Rowell, and Witherspoon: Are the images of mapping class group representations associated to the modular category Mod-D^w (G) always finite? We answer this 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 ɯ is isomorphic to a Turaev-Viro-Barrett-Westbury (TVBW) representation associated to the spherical fusion category Vecw/G 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 Vecw/G-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.