Skip to main content
MathOverflow

Return to Question

formatting, added tag
Source Link
YCor
YCor
  • 63.9k
  • 5
  • 187
  • 286

Hyperbolic homology spheres with infinite $SL_2$\mathrm{SL}_2(\mathbb{C})$ character variety

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

Hyperbolic homology spheres with infinite $SL_2(\mathbb{C})$ character variety

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

$\DeclareMathOperator\SL{SL}$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.

Source Link
Renaud Detcherry
Renaud Detcherry
  • 492
  • 4
  • 6

Hyperbolic homology spheres with infinite $SL_2(\mathbb{C})$ character variety

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.