How do you figure out what hyperbolic tilings are compatible with a given surface?

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By the uniformization theorem, any Riemann surface $H$ is conformally equivalent to one of constant curvature. Let's say that $H$ is conformally equivalent to a surface with curvature $-1$.

Therefore, the hyperbolic plane is a conformal universal cover of $H$. For any tiling on $H$, there is a corresponding tiling in the hyperbolic plane.

My question is, how do you figure out for which tilings in the hyperbolic plane there exists a tiling in $H$ that corresponds to it.

EDIT: For example, perhaps your surface is made by sewing together polygons, and the tiling in question is a m,n tiling. (This in particular makes the problem a computational decision problem.)

riemannian-geometry hyperbolic-geometry covering-spaces conformal-geometry tiling

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edited Jul 24 at 22:45

asked Jul 17 at 14:31

PyRulez

4,51822068

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  Abstract answer: When the tiling is invariant under the (hyperbolic) covering group. But maybe you have a more specific situation in mind?
  – WimC
  Jul 17 at 15:46
  

  
  
  
  


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  @WimC I was thinking of a sphere with 4 disks removed in particular.
  – PyRulez
  Jul 17 at 15:47
  

  
  
  
  


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  The abstract answer applies to your special case without change.
  – Lee Mosher
  Jul 17 at 16:20
  

  
  
  
  


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  1
  

  
  
  
  
  
  

  
  It would help a lot if you could add more clarity and explicitness to your question. Asking "how do you figure out" is not very explicit. But I'll try to give more details. By basic covering space theory, the covering group is isomorphic to the fundamental group of $H$. Furthermore, if you represent $H$ by a particular polygon with sides identified in pairs, you get an explicit presentation of the fundamental group, and an explicit embedding of that polygon as a fundamental domain of the covering group with explicit side pairing isometries.
  – Lee Mosher
  Jul 17 at 21:47
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I suggest that you take a look at the relationship between covering space theory and the fundamental group, with a particular emphasis on learning the isomorphism between the fundamental group and the covering transformation group (deck transformation group). Without knowing that, my comments (and any answer) won't be of much help to you.
  – Lee Mosher
  Jul 18 at 2:07
  

  
  
  
  

 | 
show 2 more comments

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By the uniformization theorem, any Riemann surface $H$ is conformally equivalent to one of constant curvature. Let's say that $H$ is conformally equivalent to a surface with curvature $-1$.

Therefore, the hyperbolic plane is a conformal universal cover of $H$. For any tiling on $H$, there is a corresponding tiling in the hyperbolic plane.

My question is, how do you figure out for which tilings in the hyperbolic plane there exists a tiling in $H$ that corresponds to it.

EDIT: For example, perhaps your surface is made by sewing together polygons, and the tiling in question is a m,n tiling. (This in particular makes the problem a computational decision problem.)

riemannian-geometry hyperbolic-geometry covering-spaces conformal-geometry tiling

share|cite|improve this question

edited Jul 24 at 22:45

asked Jul 17 at 14:31

PyRulez

4,51822068

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Abstract answer: When the tiling is invariant under the (hyperbolic) covering group. But maybe you have a more specific situation in mind?
  – WimC
  Jul 17 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @WimC I was thinking of a sphere with 4 disks removed in particular.
  – PyRulez
  Jul 17 at 15:47
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The abstract answer applies to your special case without change.
  – Lee Mosher
  Jul 17 at 16:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It would help a lot if you could add more clarity and explicitness to your question. Asking "how do you figure out" is not very explicit. But I'll try to give more details. By basic covering space theory, the covering group is isomorphic to the fundamental group of $H$. Furthermore, if you represent $H$ by a particular polygon with sides identified in pairs, you get an explicit presentation of the fundamental group, and an explicit embedding of that polygon as a fundamental domain of the covering group with explicit side pairing isometries.
  – Lee Mosher
  Jul 17 at 21:47
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I suggest that you take a look at the relationship between covering space theory and the fundamental group, with a particular emphasis on learning the isomorphism between the fundamental group and the covering transformation group (deck transformation group). Without knowing that, my comments (and any answer) won't be of much help to you.
  – Lee Mosher
  Jul 18 at 2:07
  

  
  
  
  

 | 
show 2 more comments

up vote
1
down vote

favorite

up vote
1
down vote

favorite

By the uniformization theorem, any Riemann surface $H$ is conformally equivalent to one of constant curvature. Let's say that $H$ is conformally equivalent to a surface with curvature $-1$.

Therefore, the hyperbolic plane is a conformal universal cover of $H$. For any tiling on $H$, there is a corresponding tiling in the hyperbolic plane.

My question is, how do you figure out for which tilings in the hyperbolic plane there exists a tiling in $H$ that corresponds to it.

EDIT: For example, perhaps your surface is made by sewing together polygons, and the tiling in question is a m,n tiling. (This in particular makes the problem a computational decision problem.)

riemannian-geometry hyperbolic-geometry covering-spaces conformal-geometry tiling

share|cite|improve this question

edited Jul 24 at 22:45

asked Jul 17 at 14:31

PyRulez

4,51822068

By the uniformization theorem, any Riemann surface $H$ is conformally equivalent to one of constant curvature. Let's say that $H$ is conformally equivalent to a surface with curvature $-1$.

Therefore, the hyperbolic plane is a conformal universal cover of $H$. For any tiling on $H$, there is a corresponding tiling in the hyperbolic plane.

My question is, how do you figure out for which tilings in the hyperbolic plane there exists a tiling in $H$ that corresponds to it.

EDIT: For example, perhaps your surface is made by sewing together polygons, and the tiling in question is a m,n tiling. (This in particular makes the problem a computational decision problem.)

riemannian-geometry hyperbolic-geometry covering-spaces conformal-geometry tiling

share|cite|improve this question

edited Jul 24 at 22:45

asked Jul 17 at 14:31

PyRulez

4,51822068

share|cite|improve this question

share|cite|improve this question

edited Jul 24 at 22:45

edited Jul 24 at 22:45

edited Jul 24 at 22:45

asked Jul 17 at 14:31

PyRulez

4,51822068

asked Jul 17 at 14:31

PyRulez

4,51822068

asked Jul 17 at 14:31

PyRulez

4,51822068

4,51822068

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Abstract answer: When the tiling is invariant under the (hyperbolic) covering group. But maybe you have a more specific situation in mind?
  – WimC
  Jul 17 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @WimC I was thinking of a sphere with 4 disks removed in particular.
  – PyRulez
  Jul 17 at 15:47
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The abstract answer applies to your special case without change.
  – Lee Mosher
  Jul 17 at 16:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It would help a lot if you could add more clarity and explicitness to your question. Asking "how do you figure out" is not very explicit. But I'll try to give more details. By basic covering space theory, the covering group is isomorphic to the fundamental group of $H$. Furthermore, if you represent $H$ by a particular polygon with sides identified in pairs, you get an explicit presentation of the fundamental group, and an explicit embedding of that polygon as a fundamental domain of the covering group with explicit side pairing isometries.
  – Lee Mosher
  Jul 17 at 21:47
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I suggest that you take a look at the relationship between covering space theory and the fundamental group, with a particular emphasis on learning the isomorphism between the fundamental group and the covering transformation group (deck transformation group). Without knowing that, my comments (and any answer) won't be of much help to you.
  – Lee Mosher
  Jul 18 at 2:07
  

  
  
  
  

 | 
show 2 more comments

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  Abstract answer: When the tiling is invariant under the (hyperbolic) covering group. But maybe you have a more specific situation in mind?
  – WimC
  Jul 17 at 15:46
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  @WimC I was thinking of a sphere with 4 disks removed in particular.
  – PyRulez
  Jul 17 at 15:47
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  The abstract answer applies to your special case without change.
  – Lee Mosher
  Jul 17 at 16:20
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  It would help a lot if you could add more clarity and explicitness to your question. Asking "how do you figure out" is not very explicit. But I'll try to give more details. By basic covering space theory, the covering group is isomorphic to the fundamental group of $H$. Furthermore, if you represent $H$ by a particular polygon with sides identified in pairs, you get an explicit presentation of the fundamental group, and an explicit embedding of that polygon as a fundamental domain of the covering group with explicit side pairing isometries.
  – Lee Mosher
  Jul 17 at 21:47
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  I suggest that you take a look at the relationship between covering space theory and the fundamental group, with a particular emphasis on learning the isomorphism between the fundamental group and the covering transformation group (deck transformation group). Without knowing that, my comments (and any answer) won't be of much help to you.
  – Lee Mosher
  Jul 18 at 2:07
  

  
  
  
  

1

1

Abstract answer: When the tiling is invariant under the (hyperbolic) covering group. But maybe you have a more specific situation in mind?
– WimC
Jul 17 at 15:46

Abstract answer: When the tiling is invariant under the (hyperbolic) covering group. But maybe you have a more specific situation in mind?
– WimC
Jul 17 at 15:46

@WimC I was thinking of a sphere with 4 disks removed in particular.
– PyRulez
Jul 17 at 15:47

@WimC I was thinking of a sphere with 4 disks removed in particular.
– PyRulez
Jul 17 at 15:47

The abstract answer applies to your special case without change.
– Lee Mosher
Jul 17 at 16:20

The abstract answer applies to your special case without change.
– Lee Mosher
Jul 17 at 16:20

1

1

It would help a lot if you could add more clarity and explicitness to your question. Asking "how do you figure out" is not very explicit. But I'll try to give more details. By basic covering space theory, the covering group is isomorphic to the fundamental group of $H$. Furthermore, if you represent $H$ by a particular polygon with sides identified in pairs, you get an explicit presentation of the fundamental group, and an explicit embedding of that polygon as a fundamental domain of the covering group with explicit side pairing isometries.
– Lee Mosher
Jul 17 at 21:47

It would help a lot if you could add more clarity and explicitness to your question. Asking "how do you figure out" is not very explicit. But I'll try to give more details. By basic covering space theory, the covering group is isomorphic to the fundamental group of $H$. Furthermore, if you represent $H$ by a particular polygon with sides identified in pairs, you get an explicit presentation of the fundamental group, and an explicit embedding of that polygon as a fundamental domain of the covering group with explicit side pairing isometries.
– Lee Mosher
Jul 17 at 21:47

1

1

I suggest that you take a look at the relationship between covering space theory and the fundamental group, with a particular emphasis on learning the isomorphism between the fundamental group and the covering transformation group (deck transformation group). Without knowing that, my comments (and any answer) won't be of much help to you.
– Lee Mosher
Jul 18 at 2:07

I suggest that you take a look at the relationship between covering space theory and the fundamental group, with a particular emphasis on learning the isomorphism between the fundamental group and the covering transformation group (deck transformation group). Without knowing that, my comments (and any answer) won't be of much help to you.
– Lee Mosher
Jul 18 at 2:07

 | 
show 2 more comments

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