$\DeclareMathOperator\SL{SL}$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.



Hyperbolic homology spheres with infinite $\mathrm{SL}_2(\mathbb{C})$ character variety
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.


