Which 3-manifolds can be cubulated?
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I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with boundary embedded in $mathbb{R}^3$, but would be happy to hear any answers to this question. If I had to name one concrete question, it would be:
Question: Can you cubulate every compact 3-manifold $M subset mathbb{R}^3$? If not, which ones can you cubulate? What are some specific examples of non-cubulable $M$?
I am aware of some scattered results, e.g.,
- all hyperbolic 3-manifolds are cubulable (discussed in Sections 4.5, 4.6 of this paper)
- there is some discussion about cubulability of Kähler groups/Kähler manifolds here
- there is a characterization in terms of the boundary here
Apart from the result about hyperbolic 3-manifolds, I find it quite hard to connect largely algebraic results like these back to a more concrete geometric/topological picture.
differential-geometry algebraic-topology differential-topology geometric-topology geometric-group-theory
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
$endgroup$
add a comment |
$begingroup$
I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with boundary embedded in $mathbb{R}^3$, but would be happy to hear any answers to this question. If I had to name one concrete question, it would be:
Question: Can you cubulate every compact 3-manifold $M subset mathbb{R}^3$? If not, which ones can you cubulate? What are some specific examples of non-cubulable $M$?
I am aware of some scattered results, e.g.,
- all hyperbolic 3-manifolds are cubulable (discussed in Sections 4.5, 4.6 of this paper)
- there is some discussion about cubulability of Kähler groups/Kähler manifolds here
- there is a characterization in terms of the boundary here
Apart from the result about hyperbolic 3-manifolds, I find it quite hard to connect largely algebraic results like these back to a more concrete geometric/topological picture.
differential-geometry algebraic-topology differential-topology geometric-topology geometric-group-theory
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
$endgroup$
$begingroup$
Graph manifolds which are cubulated have a characterization in this paper by Hagen and Przytycki
$endgroup$
– Paul Plummer
Dec 12 '18 at 20:39
4
$begingroup$
Note that the original meaning of "cubulated" was that there's a cubulation of the manifold in the proper sense (namely that there exists a locally CAT(0) cube complex that is homeomorphic to the manifold). The meaning in the Hagen-Przytycki is weaker: it means that the fundamental group acts properly cocompactly on a CAT(0) cube complex (which can be of much higher dimension). Saying that the manifold is cubulable to mean this is a bit misleading.
$endgroup$
– YCor
Dec 12 '18 at 23:33
$begingroup$
@YCor Thank you! That in itself is a very valuable clarification.
$endgroup$
– JacquesMartin
Dec 13 '18 at 0:37
add a comment |
$begingroup$
I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with boundary embedded in $mathbb{R}^3$, but would be happy to hear any answers to this question. If I had to name one concrete question, it would be:
Question: Can you cubulate every compact 3-manifold $M subset mathbb{R}^3$? If not, which ones can you cubulate? What are some specific examples of non-cubulable $M$?
I am aware of some scattered results, e.g.,
- all hyperbolic 3-manifolds are cubulable (discussed in Sections 4.5, 4.6 of this paper)
- there is some discussion about cubulability of Kähler groups/Kähler manifolds here
- there is a characterization in terms of the boundary here
Apart from the result about hyperbolic 3-manifolds, I find it quite hard to connect largely algebraic results like these back to a more concrete geometric/topological picture.
differential-geometry algebraic-topology differential-topology geometric-topology geometric-group-theory
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
$endgroup$
I am trying to get a picture of what is currently known about cubulability of 3-manifolds, though cannot seem to find a good overview. I am personally most interested in compact 3-manifolds with boundary embedded in $mathbb{R}^3$, but would be happy to hear any answers to this question. If I had to name one concrete question, it would be:
Question: Can you cubulate every compact 3-manifold $M subset mathbb{R}^3$? If not, which ones can you cubulate? What are some specific examples of non-cubulable $M$?
I am aware of some scattered results, e.g.,
- all hyperbolic 3-manifolds are cubulable (discussed in Sections 4.5, 4.6 of this paper)
- there is some discussion about cubulability of Kähler groups/Kähler manifolds here
- there is a characterization in terms of the boundary here
Apart from the result about hyperbolic 3-manifolds, I find it quite hard to connect largely algebraic results like these back to a more concrete geometric/topological picture.
differential-geometry algebraic-topology differential-topology geometric-topology geometric-group-theory
differential-geometry algebraic-topology differential-topology geometric-topology geometric-group-theory
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
edited Dec 12 '18 at 14:05
Paul Plummer
5,29221950
5,29221950
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
asked Dec 12 '18 at 13:48
JacquesMartinJacquesMartin
784
784
$begingroup$
Graph manifolds which are cubulated have a characterization in this paper by Hagen and Przytycki
$endgroup$
– Paul Plummer
Dec 12 '18 at 20:39
4
$begingroup$
Note that the original meaning of "cubulated" was that there's a cubulation of the manifold in the proper sense (namely that there exists a locally CAT(0) cube complex that is homeomorphic to the manifold). The meaning in the Hagen-Przytycki is weaker: it means that the fundamental group acts properly cocompactly on a CAT(0) cube complex (which can be of much higher dimension). Saying that the manifold is cubulable to mean this is a bit misleading.
$endgroup$
– YCor
Dec 12 '18 at 23:33
$begingroup$
@YCor Thank you! That in itself is a very valuable clarification.
$endgroup$
– JacquesMartin
Dec 13 '18 at 0:37
add a comment |
$begingroup$
Graph manifolds which are cubulated have a characterization in this paper by Hagen and Przytycki
$endgroup$
– Paul Plummer
Dec 12 '18 at 20:39
4
$begingroup$
Note that the original meaning of "cubulated" was that there's a cubulation of the manifold in the proper sense (namely that there exists a locally CAT(0) cube complex that is homeomorphic to the manifold). The meaning in the Hagen-Przytycki is weaker: it means that the fundamental group acts properly cocompactly on a CAT(0) cube complex (which can be of much higher dimension). Saying that the manifold is cubulable to mean this is a bit misleading.
$endgroup$
– YCor
Dec 12 '18 at 23:33
$begingroup$
@YCor Thank you! That in itself is a very valuable clarification.
$endgroup$
– JacquesMartin
Dec 13 '18 at 0:37
$begingroup$
Graph manifolds which are cubulated have a characterization in this paper by Hagen and Przytycki
$endgroup$
– Paul Plummer
Dec 12 '18 at 20:39
$begingroup$
Graph manifolds which are cubulated have a characterization in this paper by Hagen and Przytycki
$endgroup$
– Paul Plummer
Dec 12 '18 at 20:39
4
4
$begingroup$
Note that the original meaning of "cubulated" was that there's a cubulation of the manifold in the proper sense (namely that there exists a locally CAT(0) cube complex that is homeomorphic to the manifold). The meaning in the Hagen-Przytycki is weaker: it means that the fundamental group acts properly cocompactly on a CAT(0) cube complex (which can be of much higher dimension). Saying that the manifold is cubulable to mean this is a bit misleading.
$endgroup$
– YCor
Dec 12 '18 at 23:33
$begingroup$
Note that the original meaning of "cubulated" was that there's a cubulation of the manifold in the proper sense (namely that there exists a locally CAT(0) cube complex that is homeomorphic to the manifold). The meaning in the Hagen-Przytycki is weaker: it means that the fundamental group acts properly cocompactly on a CAT(0) cube complex (which can be of much higher dimension). Saying that the manifold is cubulable to mean this is a bit misleading.
$endgroup$
– YCor
Dec 12 '18 at 23:33
$begingroup$
@YCor Thank you! That in itself is a very valuable clarification.
$endgroup$
– JacquesMartin
Dec 13 '18 at 0:37
$begingroup$
@YCor Thank you! That in itself is a very valuable clarification.
$endgroup$
– JacquesMartin
Dec 13 '18 at 0:37
add a comment |
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$begingroup$
Graph manifolds which are cubulated have a characterization in this paper by Hagen and Przytycki
$endgroup$
– Paul Plummer
Dec 12 '18 at 20:39
4
$begingroup$
Note that the original meaning of "cubulated" was that there's a cubulation of the manifold in the proper sense (namely that there exists a locally CAT(0) cube complex that is homeomorphic to the manifold). The meaning in the Hagen-Przytycki is weaker: it means that the fundamental group acts properly cocompactly on a CAT(0) cube complex (which can be of much higher dimension). Saying that the manifold is cubulable to mean this is a bit misleading.
$endgroup$
– YCor
Dec 12 '18 at 23:33
$begingroup$
@YCor Thank you! That in itself is a very valuable clarification.
$endgroup$
– JacquesMartin
Dec 13 '18 at 0:37