Left-orderability and SU(2)-representations

Problem: Prove that if Y is an irreducible 3-manifold with left-orderable fundamental group then there exists a homomorphism \pi_1(Y)\to SU(2) with nonabelian image.

Suppose Y is irreducible with left-orderable fundamental group. Then the L-space conjecture predicts that Y is not an L-space. Together with Kronheimer-Mrowka's conjectured isomorphism \widehat{\mathit{HF}}(Y;\mathbb{C}) \cong_{\mathrm{Conj.}} I^\sharp(Y;\mathbb{C}) between Heegaard Floer homology and framed instanton homology, this would imply that Y is not an instanton L-space. If Y satisfies a further mild non-degeneracy condition (and perhaps even if Y doesn't) then being an instanton non-L-space implies that there exists a homomorphism \pi_1(Y)\to SU(2) with nonabelian image, per Baldwin-Sivek. While these conjectures serve as motivation, one can try to prove the stated link between left-orderability and nonabelian SU(2)-representations directly. That's the goal of this problem.

The left-orderability of \pi_1(Y) is equivalent to the existence of a nontrivial homomorphism \pi_1(Y) \to \mathrm{Homeo}^+(\mathbb{R}), by Boyer-Rolfsen-Wiest. If Y is a homology sphere in addition to being irreducible, then the existence of such a homomorphism is equivalent to the existence of a nontrivial homomorphism \pi_1(Y) \to \mathrm{Homeo}^+(S^1), since the first clearly implies the second, and the obstruction to lifting the second to the first lies in H^2(Y) = 0. As there is always a nontrivial homomorphism \pi_1(Y)\to SU(2) when Y is not a homology sphere, it follows that a solution to the problem at the top would imply a solution to the following:

Problem: Suppose that Y is an irreducible 3-manifold. Prove that if there exists a nontrivial homomorphism \pi_1(Y) \to \mathrm{Homeo}^+(S^1) then there exists a nontrivial homomorphism \pi_1(Y) \to SU(2).

In this vein, we remark that there exists a nontrivial homomorphism \pi_1(Y) \to \mathrm{Homeo}^+(S^1) whenever Y admits a coorientable taut foliation, by Calegari-Dunfield. In this case, we also know that there is a nontrivial homomorphism \pi_1(Y) \to SU(2) via instanton Floer methods.

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