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Hyperbolic 3-manifolds of finite volume as link complements

1

1

$begingroup$

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference).

Let $N$ be a hyperbolic 3-manifold of finite volume and let $N_1,,N_2,,ldots,,N_k$ be pairwise disjoint representatives of the cusp ends of $N$, so each $N_i$ is diffeomorphic to $mathbbT^2times [0,infty)$. Let $M = Nsetminus (N_1cupldotscup N_k)$. Then, $M$ is a compact 3-manifold with incompressible toroidal boundary.

1) After performing Dehn filling on the cusp ends of $N$ so that the resulting closed manifold $widehatN$ is hyperbolic, is there link $Gammasubset widehatN$ such that $N$ is diffeomorphic to $widehatNsetminus Gamma$?

2) Assuming 1) is true, how different Dehn fillings on the above procedure change $widehatN$ and $Gamma$ as above? In other words, if we produce $widehatN_1$ and $widehatN_2$ trough distinct Dehn fillings in $N$ are $widehatN_1$ and $widehatN_2$ diffeomorphic? If they are, are the links $Gamma_1$ and $Gamma_2$ isotopic?

I am trying to understand (say orientable) hyperbolic 3-manifolds (of finite volume) and the standard examples are link complements in a closed three manifold, so this question is more or less asking if for a given hyperbolic 3-manifold of finite volume there is only one way of seeing it topologically as a link complement. Thank you.

hyperbolic-geometry geometric-topology low-dimensional-topology

share|cite|improve this question

asked Mar 18 at 21:58

LLimaLLima

404

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  1 is obviously true. Part 2 is somewhat true and somewhat false. What is true is that "most" Dehn fillings will be non-isometric, one can see this by looking at the hyperbolic volume. What papers are you reading?
  $endgroup$
  – Moishe Kohan
  Mar 19 at 23:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sure! Got it! Thanks @MoisheKohan. I am almost randomly reading Thurston's notes and the "3-manifold groups" book by Aschenbrenner, Friedl and Wilton. After this question, I think things are more clear in my mind.
  $endgroup$
  – LLima
  Mar 21 at 22:09
  

  
  
  
  

add a comment | 

1

1

$begingroup$

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference).

Let $N$ be a hyperbolic 3-manifold of finite volume and let $N_1,,N_2,,ldots,,N_k$ be pairwise disjoint representatives of the cusp ends of $N$, so each $N_i$ is diffeomorphic to $mathbbT^2times [0,infty)$. Let $M = Nsetminus (N_1cupldotscup N_k)$. Then, $M$ is a compact 3-manifold with incompressible toroidal boundary.

1) After performing Dehn filling on the cusp ends of $N$ so that the resulting closed manifold $widehatN$ is hyperbolic, is there link $Gammasubset widehatN$ such that $N$ is diffeomorphic to $widehatNsetminus Gamma$?

2) Assuming 1) is true, how different Dehn fillings on the above procedure change $widehatN$ and $Gamma$ as above? In other words, if we produce $widehatN_1$ and $widehatN_2$ trough distinct Dehn fillings in $N$ are $widehatN_1$ and $widehatN_2$ diffeomorphic? If they are, are the links $Gamma_1$ and $Gamma_2$ isotopic?

I am trying to understand (say orientable) hyperbolic 3-manifolds (of finite volume) and the standard examples are link complements in a closed three manifold, so this question is more or less asking if for a given hyperbolic 3-manifold of finite volume there is only one way of seeing it topologically as a link complement. Thank you.

hyperbolic-geometry geometric-topology low-dimensional-topology

share|cite|improve this question

asked Mar 18 at 21:58

LLimaLLima

404

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  1 is obviously true. Part 2 is somewhat true and somewhat false. What is true is that "most" Dehn fillings will be non-isometric, one can see this by looking at the hyperbolic volume. What papers are you reading?
  $endgroup$
  – Moishe Kohan
  Mar 19 at 23:44
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Sure! Got it! Thanks @MoisheKohan. I am almost randomly reading Thurston's notes and the "3-manifold groups" book by Aschenbrenner, Friedl and Wilton. After this question, I think things are more clear in my mind.
  $endgroup$
  – LLima
  Mar 21 at 22:09
  

  
  
  
  

add a comment | 

$begingroup$

This is an improved version of this question (sorry if I wasn't so clear there and sorry if this is well-known and I didn't find the reference).

Let $N$ be a hyperbolic 3-manifold of finite volume and let $N_1,,N_2,,ldots,,N_k$ be pairwise disjoint representatives of the cusp ends of $N$, so each $N_i$ is diffeomorphic to $mathbbT^2times [0,infty)$. Let $M = Nsetminus (N_1cupldotscup N_k)$. Then, $M$ is a compact 3-manifold with incompressible toroidal boundary.

1) After performing Dehn filling on the cusp ends of $N$ so that the resulting closed manifold $widehatN$ is hyperbolic, is there link $Gammasubset widehatN$ such that $N$ is diffeomorphic to $widehatNsetminus Gamma$?

2) Assuming 1) is true, how different Dehn fillings on the above procedure change $widehatN$ and $Gamma$ as above? In other words, if we produce $widehatN_1$ and $widehatN_2$ trough distinct Dehn fillings in $N$ are $widehatN_1$ and $widehatN_2$ diffeomorphic? If they are, are the links $Gamma_1$ and $Gamma_2$ isotopic?

I am trying to understand (say orientable) hyperbolic 3-manifolds (of finite volume) and the standard examples are link complements in a closed three manifold, so this question is more or less asking if for a given hyperbolic 3-manifold of finite volume there is only one way of seeing it topologically as a link complement. Thank you.

hyperbolic-geometry geometric-topology low-dimensional-topology

share|cite|improve this question

asked Mar 18 at 21:58

LLimaLLima

404

$endgroup$

share|cite|improve this question

asked Mar 18 at 21:58

LLimaLLima

404

share|cite|improve this question

asked Mar 18 at 21:58

LLimaLLima

404

asked Mar 18 at 21:58

LLimaLLima

404

asked Mar 18 at 21:58

LLimaLLima

404