  We show that any twisted Dijkgraaf-Witten representation of a mapping class
group of an orientable, compact surface with boundary has finite image. This
generalizes work of Etingof, Rowell and Witherspoon showing that the braid
group images are finite. In particular, our result answers their question
regarding finiteness of images of arbitrary mapping class group representations
in the affirmative.
  Our approach is to translate the problem into manipulation of colored graphs
embedded in the given surface. To do this translation, we use the fact that any
twisted Dijkgraaf-Witten 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.
