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$\DeclareMathOperator\SL{SL}$By the celebrated results of Culler and Shalen, a closed $3$-manifold contains an incompressible surface if its $\SL_2(\mathbb{C})$ character variety is infinite.

Now, for manifolds with positive $b_1,$ there is a much easier argument for existence of incompressible surfaces. In that regard, Culler-Shalen theory looks more striking when applied to rational homology spheres. Also maybe it is more interesting to look at hyperbolic homology spheres, since it is really surfaces of higher genera that we would like to detect.

Now my question is that, looking at the litterature, I was not able to find an example of a rational hyperbolic homology sphere with infinite $\SL_2(\mathbb{C})$ character variety. I am sure they exist though, so I would like to see if anyone has a nice example.

I would guess that the surgery on a knot along a boundary slope corresponding to a genus $g\geq 2$ surface would be a good candidate, but I have no idea whether those would typically have infinite $\SL_2(\mathbb{C})$ character variety.

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Take two knot complements (say of $K,K'$) and glue them together, interchanging meridians and longitudes. This is called splicing and produces a homology sphere $S(K,K')$; if both knots are non-trivial then the boundary torus where you glued is incompressible. It's standard that this produces a higher dimensional component of the character variety. See eg Boden-Curtis, Splicing and the SL2(C) Casson invariant, Proc. Amer. Math. Soc. 136 (2008), no. 7, 2615–2623. You don't need any Culler-Shalen theory for this (see Theorem 3.1).

Because of the incompressible torus, $S(K,K')$ is not hyperbolic. But there is always a hyperbolic homology sphere $Y$ with a degree one map $Y \to S(K,K')$. Then the character variety of $Y$ will have $\dim > 0$ as well. There are variations/elaborations on this theme, eg you could assume that there is an invertible homology cobordism from $S(K,K')$ to $Y$ which induces the degree one map.

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    $\begingroup$ Could you provide a reference for the construction of the hyperbolic homology sphere that $\pi_1$-surjects the spliced manifold? $\endgroup$
    – HJRW
    Commented Mar 16, 2023 at 9:55
  • $\begingroup$ Drill out a curve so the complement is hyperbolic, then do a hyperbolic Dehn filling. The filled manifold admits a degree $1$ map to $S(K, K^\prime)$. $\endgroup$
    – Toffee
    Commented Mar 16, 2023 at 12:50
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    $\begingroup$ @Toffee I don't see why the Dehn filled manifold admits a degree one map to the original manifold. You'd have to know that the map on the complement of your hyperbolic knot extends over the torus you glue in. If the knot is null-homotopic, there is an obstruction theory argument for this. But I don't see why there would be a null homotopic curve with that property. $\endgroup$
    – Danny Ruberman
    Commented Mar 17, 2023 at 4:35
  • $\begingroup$ @HJRW One construction of a degree one map comes from invertible homology cobordisms. I constructed such cobordisms and maps in an old paper, Seifert surfaces of knots in $S^4$. Pacific J. Math. 145 (1990), no. 1, 97–116. $\endgroup$
    – Danny Ruberman
    Commented Mar 17, 2023 at 4:38
  • $\begingroup$ Yes, you do want it null-homotopic for the extension, and I agree in retrospect this is dodgy for a splicing. $\endgroup$
    – Toffee
    Commented Mar 17, 2023 at 12:27

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