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!



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 $SL(2,\mathbb{C})$ (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](http://www.ams.org/mathscinet-getitem?mr=1937957) 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](http://www.ams.org/mathscinet-getitem?mr=1433117). 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](http://www.ams.org/mathscinet-getitem?mr=2106239) 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](http://www.ams.org/mathscinet-getitem?mr=1459136), 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. 


