Any twisted Dijkgraaf-Witten representation of a mapping class group of an orientable, compact surface with boundary has finite image.
Prior to the current work, certain specific cases had already been solved. In the case of the torus, Ng and Schauenburg's Congruence Subgroup Theorem implies the much stronger result that any Reshitikhin-Turaev representation of the mapping class group of the torus has finite image @cite_1 . Another related result is due to Fjelstad and Fuchs @cite_3 . They showed that, given a surface with at most one boundary component, the mapping class group representations corresponding to the untwisted (i.e. @math ) Dijkgraaf-Witten theory have finite image. Their paper uses an algebraic method of Lyubashenko @cite_7 that gives a projective mapping class group representation to any factorizable ribbon Hopf algebra, in their case, the double @math . In our case, we instead consider the mapping class group action on a vector space of @math -colored embedded graphs defined by Kirillov @cite_13 , yielding a simpler, geometric proof of the more general twisted case.
@cite_13 @cite_1 @cite_7 @cite_3
567868760 1964155237 2952236103 2147668104
We investigate representations of mapping class groups of surfaces that arise from the untwisted Drinfeld double of a finite group G, focusing on surfaces without marked points or with one marked point. We obtain concrete descriptions of such representations in terms of finite group data. This allows us to establish various properties of these representations. In particular we show that they have finite images, and that for surfaces of genus at least 3 their restriction to the Torelli group is non-trivial iff G is non-abelian. We conjecture that every finite groupGhas a short presentation (in terms of generators and relations) in the sense that the totallengthof the relations is (log|G|)O(1). We show that it suffices to prove this conjecture for simple groups. Motivated by applications in computational complexity theory, we conjecture that for finite simple groups, such a short presentation is computable in polynomial time from the standard name ofG, assuming in the case of Lie type simple groups overGF(pm) that an irreducible polynomialfof degreemoverGF(p) and a primitive root ofGF(pm) are given. We verify this (stronger) conjecture for all finite simple groups except for the three families of rank 1 twisted groups: we do not handle the unitary groupsPSU(3, q) = 2A2(q), the Suzuki groupsSz(q) = 2B2(q), and the Ree groupsR(q) = 2G2(q). In particular,all finite groups G without composition factors of these types have presentations of length O((log|G|)3). For groups of Lie type (normal or twisted) of rank ≥ 2, we use a reduced version of the Curtis–Steinberg–Tits presentation. In the Cayley graph of the mapping class group of a closed surface, with respect to any generating set, we look at a ball of large radius centered on the identity vertex, and at the proportion among the vertices in this ball representing pseudo-Anosov elements. A well-known conjecture states that this proportion should tend to one as the radius tends to infinity. We prove that it stays bounded away from zero. We also prove similar results for a large class of subgroups of the mapping class group. We show that the Goemans-Linial semidefinite relaxation of the Sparsest Cut problem with general demands has integrality gap @math . This is achieved by exhibiting @math -point metric spaces of negative type whose @math distortion is @math . Our result is based on quantitative bounds on the rate of degeneration of Lipschitz maps from the Heisenberg group to @math when restricted to cosets of the center.