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Can someone give me a non-trivial example of a flat SU(2)-connection over a compact orientable hyperbolic 3-manifold?

The literature on such bundles over 3-manifolds is huge and my naive searches don't seem to turn up specific examples.

Roughly speaking, the Casson invariant counts flat bundles over 3-manifolds, so in principal I suppose I would be happy with an example of a hyperbolic 3-manifold with non-zero SU(2) Casson invariant (and surely such things are known). In practice, I would really like to see the non-trivial bundle (or corresponding representation) more-or-less explicitly.

Finally, I would also be happy with just being told precisely where I should go and look in the literature!

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3 Answers 3

up vote 15 down vote accepted

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 $SL(2,\mathbb{C})$ (lifted from $PSL(2,\mathbb{C})$) may be conjugated so that it lies in $SL(2,K)$, for $K$ a number field (because transcendental extensions have infinitesimal deformations in $\mathbb{C}$). In particular, the traces of elements will always lie in a number field. One may take different Galois embeddings of $K$ into $\mathbb{C}$, and get new representations of $\Gamma$ into $SL(2,\mathbb{C})$. 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 $SL(2,\mathbb{R})$ 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 $p\pi/q$, and get a representation into $O(4)$. Passing to finite index manifold subgroups, one can obtain $SU(2)$ reps. (since $SO(4)$ is essentially $SU(2)\times SU(2)$).

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 $\mathbb{Z}/2$ extension of $A_5$) 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 $A_5 < SO(3)$, 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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Thank you very much, that's a detailed answer and I really appreciate the references also. The picture you describe for hyperbolic reflection groups looks particularly explicit. –  Joel Fine Dec 15 '09 at 10:19

The figure eight knot is hyperbolic, so by Thurston all but finitely many 1/n-surgeries on it yield hyperbolic homology spheres. The Casson invariant of the 1/n surgery is (n/2)Δ''(1), where Δ(t) = -t + 3 - t-1 is the Alexander polynomial of the figure eight, so it should be -n and therefore there are at least n distinct SU(2) representations.

In practice you should be able to work these out explicitly from a presentation of the knot group: figure out the SU(2) representation variety of the knot group (see Klassen, "Representations of knot groups in SU(2)" for the figure eight), impose the relation μλn=1, and compute what's left.

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Thanks very much. I guess that's as explicit as I could have hoped for. So you get +1, but I'll accept Agol's answer since it is more comprehensive. –  Joel Fine Dec 15 '09 at 10:15

There is a huge literature on this. I second starting with Klassen's article. You should also go back to Riley's old article and then look at Burde for 2-bridge knots. But I suspect that noone knows an "explicit" description in the sense that "this loop goes to this matrix" in general the representations are given by the real points of a variety defined over Z.

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Yes, somehow the literature is so huge that just searching "hyperbolic flat SU(2)-connection" gives so many hits that it's useless! Thanks for the references and +1 to you. –  Joel Fine Dec 15 '09 at 10:14

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