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## "Finiteness for mapping class group representations from twisted Dijkgraaf-Witten theory", Paul Gustafson, 2018

*Reviewed January 15, 2024*

*Citation:* Gustafson, Paul P. "Finiteness for mapping class group representations from twisted Dijkgraaf-Witten theory." Journal of Knot Theory and Its Ramifications 27.06 (2018): 1850043.

*Web:* https://arxiv.org/abs/1610.06069

*Tags:* Kitaev-quantum-double, Modular-tensor-categories, Topological-quantum-field-theory, No-go

In this paper, it is proved that the mapping class group representations
associated to twisted quantum doubles of finite groups all have finite image.
This includes the braid group representations as a special case, which
was already established earlier in

> P. Etingof, E. C. Rowell, and S. Witherspoon, Braid group representations from twisted quantum doubles of finite groups, Pacific J. Math. 234 (2008), no. 1, 33-42

The Property-F conjecture states that a MTC should have finite braid
group representations if and only if it is weakly integral,
and a generalization states that it should have finite braid
group representations if and only if it has finite
mapping class group representations on every surface.
This papers verifies this result in the special case
of group-theoretical topological order.

The results of this paper are very combinatorial and direct.
The proof is quite simple, and one can perhaps
imagine it as a blueprint for further progress.