Tuesday, 9 October 2012

gt.geometric topology - Flat SU(2) bundles over hyperbolic 3-manifolds

Many (compact orientable) hyperbolic 3-manifolds have non-trivial SU(2) representations.



By Mostow rigidity, the representation of the fundamental group Gamma of a closed hyperbolic 3-manifold into SL(2,mathbbC) (lifted from PSL(2,mathbbC)) may be conjugated so that it lies in SL(2,K), for K a number field (because transcendental extensions have infinitesimal deformations in mathbbC). In particular, the traces of elements will always lie in a number field. One may take different Galois embeddings of K into mathbbC, and get new representations of Gamma into SL(2,mathbbC). Sometimes, this representation is just conjugate to the original (e.g. if K was chosen too large), but in other cases the new representation of Gamma lies in SL(2,mathbbR) or in SU(2). A nice class of examples of this type are arithmetic hyperbolic 3-manifolds. In fact, they are characterized by the fact that all traces of elements are algebraic integers, and non-trivial Galois embeddings lie in SU(2) (you have to be a bit careful about what this means). Some arithmetic manifolds will only have the complex conjugate representation this way (basically, if squares of the traces lie in a quadratic imaginary number field), but otherwise you get a non-trivial SU(2) representation. The simplest example is the Weeks manifold, with trace field a cubic field. I suggest the book by MacLachlan and Reid as an introduction to arithmetic 3-manifolds. The description I've given though is encoded in terms of quaternion algebras and other algebraic machinery. Another characterization of arithmeticity is in this paper. The nice thing about these representations is that they are faithful.
There is a very explicit way to see these representations for hyperbolic reflection groups (studied by Vinberg in the arithmetic case). Basically, given a hyperbolic polyhedron with acute angles of the form pi/q, sometimes one can form a spherical polyhedron with corresponding angles which are ppi/q, and get a representation into O(4). Passing to finite index manifold subgroups, one can obtain SU(2) reps. (since SO(4) is essentially SU(2)timesSU(2)).



There are other ways one has SU(2) representations, but they are less explicit.
Kronheimer and Mrowka have shown that any non-trivial integral surgery on a knot has a non-abelian SU(2) representation. Also, any hyperbolic 3-manifold with first betti number positive or a smooth taut orientable foliation has non-abelian SU(2) representations.



Addendum: Another observation relating SU(2) representations to hyperbolic geometry is via the observation that the binary icosahedral group (a mathbbZ/2 extension of A5) is a subgroup of SU(2). By an observation of Long and Reid, every hyperbolic 3-manifold group has infinitely many quotients of the form PSL(2,p), p prime. These groups always contain subgroups isomomorphic to A5<SO(3), so one may find a finite-sheeted cover which has a non-abelian SO(3) and therefore SU(2) representation. I have no idea though whether these representations are detected by the Casson invariant or Floer homology.

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