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Answered the first question based on comments
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Shijie Gu
Shijie Gu
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Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action?

Here "properly discontinuous" doesn't have to be fixed point free, but the stabilizer of each point must be finite. The question is inspired by an interesting question asked by Agelos. I suspect

Edit: Thanks to Igor Belegradek and Ian Agol's comments, the answer is nonegative due to the (orbiford) geometrization.

Myers ON MAPPING CLASS GROUPS OF CONTRACTIBLE OPEN 3-MANIFOLDS proved that the only finite groups which can act on the Whitehead manifold is cyclic group of order 2. For genus two example the only finite groups which can act are $\mathbb{Z}_2$ and $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. Those actions are related to the involutions.

Is there any classification of finitely generated groups acting properly discontinuously and cocompactly on a simply connected open orientable 3-manifold?

Maillot Thm. 1.2 showed that finitely generated groups acting smoothly, properly discontinuously and cocompactly on an open orientable 3-manifold with infinite cyclic $\pi_1$ are virtually closed surface group. So what can one say if the manifold is simply connected?

Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action?

Here "properly discontinuous" doesn't have to be fixed point free, but the stabilizer of each point must be finite. The question is inspired by an interesting question asked by Agelos. I suspect the answer is no.

Myers ON MAPPING CLASS GROUPS OF CONTRACTIBLE OPEN 3-MANIFOLDS proved that the only finite groups which can act on the Whitehead manifold is cyclic group of order 2. For genus two example the only finite groups which can act are $\mathbb{Z}_2$ and $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. Those actions are related to the involutions.

Is there any classification of finitely generated groups acting properly discontinuously and cocompactly on a simply connected open orientable 3-manifold?

Maillot Thm. 1.2 showed that finitely generated groups acting smoothly, properly discontinuously and cocompactly on an open orientable 3-manifold with infinite cyclic $\pi_1$ are virtually closed surface group. So what can one say if the manifold is simply connected?

Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action?

Here "properly discontinuous" doesn't have to be fixed point free, but the stabilizer of each point must be finite. The question is inspired by an interesting question asked by Agelos.

Edit: Thanks to Igor Belegradek and Ian Agol's comments, the answer is negative due to the (orbiford) geometrization.

Myers ON MAPPING CLASS GROUPS OF CONTRACTIBLE OPEN 3-MANIFOLDS proved that the only finite groups which can act on the Whitehead manifold is cyclic group of order 2. For genus two example the only finite groups which can act are $\mathbb{Z}_2$ and $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. Those actions are related to the involutions.

Is there any classification of finitely generated groups acting properly discontinuously and cocompactly on a simply connected open orientable 3-manifold?

Maillot Thm. 1.2 showed that finitely generated groups acting smoothly, properly discontinuously and cocompactly on an open orientable 3-manifold with infinite cyclic $\pi_1$ are virtually closed surface group. So what can one say if the manifold is simply connected?

Source Link
Shijie Gu
Shijie Gu
  • 2.1k
  • 1
  • 15
  • 18

Can Whitehead manifold admit a properly discontinuous cocompact group action?

Can classical contractible manifolds such as Whitehead manifold admit a properly discontinuous cocompact group action?

Here "properly discontinuous" doesn't have to be fixed point free, but the stabilizer of each point must be finite. The question is inspired by an interesting question asked by Agelos. I suspect the answer is no.

Myers ON MAPPING CLASS GROUPS OF CONTRACTIBLE OPEN 3-MANIFOLDS proved that the only finite groups which can act on the Whitehead manifold is cyclic group of order 2. For genus two example the only finite groups which can act are $\mathbb{Z}_2$ and $\mathbb{Z}_2 \oplus \mathbb{Z}_2$. Those actions are related to the involutions.

Is there any classification of finitely generated groups acting properly discontinuously and cocompactly on a simply connected open orientable 3-manifold?

Maillot Thm. 1.2 showed that finitely generated groups acting smoothly, properly discontinuously and cocompactly on an open orientable 3-manifold with infinite cyclic $\pi_1$ are virtually closed surface group. So what can one say if the manifold is simply connected?