Understanding a comment by ThurstonUnderstanding the Hopf fibrationThe Birman–Hilden Theorem and the Nielsen–Thurston classificationBig picture question about the Thurston Metric on Teichmüller SpaceProblem in understanding models of hyperbolic geometryPoincaré conjecture for positively curved Thurston geometriesUnderstanding how the Thurston Geometrization conjecture implies the Poincaré conjecture.Are there many hyperbolic structures on $ I times$ Torus?Understanding cobordisms constructed from a Heegaard tripleUnderstanding a Bill Thurston popularization of knot complementsComputing Gaussian curvature of hyperbolic space using Thurston
Trouble reading roman numeral notation with flats
Air travel with refrigerated insulin
C++ lambda syntax
What is it called when someone votes for an option that's not their first choice?
What is this high flying aircraft over Pennsylvania?
Is divisi notation needed for brass or woodwind in an orchestra?
Offset in split text content
Would a primitive species be able to learn English from reading books alone?
Sort with assumptions
Is there a POSIX way to shutdown a UNIX machine?
New Order #2: Turn My Way
Highest stage count that are used one right after the other?
Did I make a mistake by ccing email to boss to others?
If the Dominion rule using their Jem'Hadar troops, why is their life expectancy so low?
Connection Between Knot Theory and Number Theory
Why do Radio Buttons not fill the entire outer circle?
Has the laser at Magurele, Romania reached a tenth of the Sun's power?
What do the positive and negative (+/-) transmit and receive pins mean on Ethernet cables?
Why can't I get pgrep output right to variable on bash script?
Travelling in US for more than 90 days
"Oh no!" in Latin
A seasonal riddle
Can a Knock spell open the door to Mordenkainen's Magnificent Mansion?
Capacitor electron flow
Understanding a comment by Thurston
Understanding the Hopf fibrationThe Birman–Hilden Theorem and the Nielsen–Thurston classificationBig picture question about the Thurston Metric on Teichmüller SpaceProblem in understanding models of hyperbolic geometryPoincaré conjecture for positively curved Thurston geometriesUnderstanding how the Thurston Geometrization conjecture implies the Poincaré conjecture.Are there many hyperbolic structures on $ I times$ Torus?Understanding cobordisms constructed from a Heegaard tripleUnderstanding a Bill Thurston popularization of knot complementsComputing Gaussian curvature of hyperbolic space using Thurston
$begingroup$
In page 359 (right after Theorem 2.3) of the following paper
Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.
W. Thurston states (and I quote)
A hyperbolic structure on the interior of a compact manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori, with the single exception of Example 2.1, which has no hyperbolic structure of finite volume.
His example 2.1 is simply the product of the 2-torus by the closed interval, $M = mathbbT^2times I$.
What did Thurston had in mind when he stated that? Why did he left the solid torus $mathbbDtimesmathbbS^1$ out of this, say, classification? I ask that to understand if it was just a blunder or if he actually had a reason to exclude this case (which can be obviously obtained by the quotient $mathbbH^3/tau$, where $tau$ is a parabolic translation in $mathbbH^3$)?
hyperbolic-geometry geometric-topology low-dimensional-topology
asked Mar 13 at 18:30
LLimaLLima
404
$endgroup$
add a comment |
$begingroup$
In page 359 (right after Theorem 2.3) of the following paper
Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.
W. Thurston states (and I quote)
A hyperbolic structure on the interior of a compact manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori, with the single exception of Example 2.1, which has no hyperbolic structure of finite volume.
His example 2.1 is simply the product of the 2-torus by the closed interval, $M = mathbbT^2times I$.
What did Thurston had in mind when he stated that? Why did he left the solid torus $mathbbDtimesmathbbS^1$ out of this, say, classification? I ask that to understand if it was just a blunder or if he actually had a reason to exclude this case (which can be obviously obtained by the quotient $mathbbH^3/tau$, where $tau$ is a parabolic translation in $mathbbH^3$)?
hyperbolic-geometry geometric-topology low-dimensional-topology
asked Mar 13 at 18:30
LLimaLLima
404
$endgroup$
$begingroup$
I think he just forgot his example. He also forgot few non-orientable examples.
$endgroup$
– Moishe Kohan
Mar 14 at 0:38
$begingroup$
I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it...
$endgroup$
– LLima
Mar 14 at 3:35
$begingroup$
I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary".
$endgroup$
– Lee Mosher
Mar 14 at 15:20
add a comment |
$begingroup$
In page 359 (right after Theorem 2.3) of the following paper
Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.
W. Thurston states (and I quote)
A hyperbolic structure on the interior of a compact manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori, with the single exception of Example 2.1, which has no hyperbolic structure of finite volume.
His example 2.1 is simply the product of the 2-torus by the closed interval, $M = mathbbT^2times I$.
What did Thurston had in mind when he stated that? Why did he left the solid torus $mathbbDtimesmathbbS^1$ out of this, say, classification? I ask that to understand if it was just a blunder or if he actually had a reason to exclude this case (which can be obviously obtained by the quotient $mathbbH^3/tau$, where $tau$ is a parabolic translation in $mathbbH^3$)?
hyperbolic-geometry geometric-topology low-dimensional-topology
asked Mar 13 at 18:30
LLimaLLima
404
$endgroup$
In page 359 (right after Theorem 2.3) of the following paper
Thurston, William P., Three dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Am. Math. Soc., New Ser. 6, 357-379 (1982). ZBL0496.57005.
W. Thurston states (and I quote)
A hyperbolic structure on the interior of a compact manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori, with the single exception of Example 2.1, which has no hyperbolic structure of finite volume.
His example 2.1 is simply the product of the 2-torus by the closed interval, $M = mathbbT^2times I$.
What did Thurston had in mind when he stated that? Why did he left the solid torus $mathbbDtimesmathbbS^1$ out of this, say, classification? I ask that to understand if it was just a blunder or if he actually had a reason to exclude this case (which can be obviously obtained by the quotient $mathbbH^3/tau$, where $tau$ is a parabolic translation in $mathbbH^3$)?
hyperbolic-geometry geometric-topology low-dimensional-topology
hyperbolic-geometry geometric-topology low-dimensional-topology
asked Mar 13 at 18:30
LLimaLLima
404
asked Mar 13 at 18:30
LLimaLLima
404
asked Mar 13 at 18:30
LLimaLLima
404
asked Mar 13 at 18:30
LLimaLLima
404
asked Mar 13 at 18:30
LLimaLLima
404
404
$begingroup$
I think he just forgot his example. He also forgot few non-orientable examples.
$endgroup$
– Moishe Kohan
Mar 14 at 0:38
$begingroup$
I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it...
$endgroup$
– LLima
Mar 14 at 3:35
$begingroup$
I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary".
$endgroup$
– Lee Mosher
Mar 14 at 15:20
add a comment |
$begingroup$
I think he just forgot his example. He also forgot few non-orientable examples.
$endgroup$
– Moishe Kohan
Mar 14 at 0:38
$begingroup$
I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it...
$endgroup$
– LLima
Mar 14 at 3:35
$begingroup$
I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary".
$endgroup$
– Lee Mosher
Mar 14 at 15:20
$begingroup$
I think he just forgot his example. He also forgot few non-orientable examples.
$endgroup$
– Moishe Kohan
Mar 14 at 0:38
$begingroup$
I think he just forgot his example. He also forgot few non-orientable examples.
$endgroup$
– Moishe Kohan
Mar 14 at 0:38
$begingroup$
I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it...
$endgroup$
– LLima
Mar 14 at 3:35
$begingroup$
I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it...
$endgroup$
– LLima
Mar 14 at 3:35
$begingroup$
I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary".
$endgroup$
– Lee Mosher
Mar 14 at 15:20
$begingroup$
I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary".
$endgroup$
– Lee Mosher
Mar 14 at 15:20
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Just to close this question. Rather than trying to analyze what Thurston was thinking when writing the statement (which is impossible without help of an ouija board) let me write the correction:
A (complete) hyperbolic structure on the interior of a compact (connected and oriented) manifold $M^3$ with incompressible boundary has finite volume if and only if $partial M^3$ consists of tori, with the exception of $T^2times I$. The latter admits no hyperbolic structure of finite volume.
If we drop the assumption of orientability and incompressible boundary then the correct statement is:
A (complete) hyperbolic structure on the interior of a compact (connected) manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori and Klein bottles, with four exceptions which are $T^2times I, K^2times I$, $D^2times S^1$ (the solid torus) and the total space of the unique nontrivial disk bundle over $S^1$ (the solid Klein bottle). The exceptional manifolds admit a hyperbolic structure but admit no hyperbolic structures of finite volume.
Here $K^2$ is the Klein bottle.
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3146985%2funderstanding-a-comment-by-thurston%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Just to close this question. Rather than trying to analyze what Thurston was thinking when writing the statement (which is impossible without help of an ouija board) let me write the correction:
A (complete) hyperbolic structure on the interior of a compact (connected and oriented) manifold $M^3$ with incompressible boundary has finite volume if and only if $partial M^3$ consists of tori, with the exception of $T^2times I$. The latter admits no hyperbolic structure of finite volume.
If we drop the assumption of orientability and incompressible boundary then the correct statement is:
A (complete) hyperbolic structure on the interior of a compact (connected) manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori and Klein bottles, with four exceptions which are $T^2times I, K^2times I$, $D^2times S^1$ (the solid torus) and the total space of the unique nontrivial disk bundle over $S^1$ (the solid Klein bottle). The exceptional manifolds admit a hyperbolic structure but admit no hyperbolic structures of finite volume.
Here $K^2$ is the Klein bottle.
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
$endgroup$
add a comment |
$begingroup$
Just to close this question. Rather than trying to analyze what Thurston was thinking when writing the statement (which is impossible without help of an ouija board) let me write the correction:
A (complete) hyperbolic structure on the interior of a compact (connected and oriented) manifold $M^3$ with incompressible boundary has finite volume if and only if $partial M^3$ consists of tori, with the exception of $T^2times I$. The latter admits no hyperbolic structure of finite volume.
If we drop the assumption of orientability and incompressible boundary then the correct statement is:
A (complete) hyperbolic structure on the interior of a compact (connected) manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori and Klein bottles, with four exceptions which are $T^2times I, K^2times I$, $D^2times S^1$ (the solid torus) and the total space of the unique nontrivial disk bundle over $S^1$ (the solid Klein bottle). The exceptional manifolds admit a hyperbolic structure but admit no hyperbolic structures of finite volume.
Here $K^2$ is the Klein bottle.
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
$endgroup$
add a comment |
$begingroup$
Just to close this question. Rather than trying to analyze what Thurston was thinking when writing the statement (which is impossible without help of an ouija board) let me write the correction:
A (complete) hyperbolic structure on the interior of a compact (connected and oriented) manifold $M^3$ with incompressible boundary has finite volume if and only if $partial M^3$ consists of tori, with the exception of $T^2times I$. The latter admits no hyperbolic structure of finite volume.
If we drop the assumption of orientability and incompressible boundary then the correct statement is:
A (complete) hyperbolic structure on the interior of a compact (connected) manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori and Klein bottles, with four exceptions which are $T^2times I, K^2times I$, $D^2times S^1$ (the solid torus) and the total space of the unique nontrivial disk bundle over $S^1$ (the solid Klein bottle). The exceptional manifolds admit a hyperbolic structure but admit no hyperbolic structures of finite volume.
Here $K^2$ is the Klein bottle.
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
$endgroup$
Just to close this question. Rather than trying to analyze what Thurston was thinking when writing the statement (which is impossible without help of an ouija board) let me write the correction:
A (complete) hyperbolic structure on the interior of a compact (connected and oriented) manifold $M^3$ with incompressible boundary has finite volume if and only if $partial M^3$ consists of tori, with the exception of $T^2times I$. The latter admits no hyperbolic structure of finite volume.
If we drop the assumption of orientability and incompressible boundary then the correct statement is:
A (complete) hyperbolic structure on the interior of a compact (connected) manifold $M^3$ has finite volume if and only if $partial M^3$ consists of tori and Klein bottles, with four exceptions which are $T^2times I, K^2times I$, $D^2times S^1$ (the solid torus) and the total space of the unique nontrivial disk bundle over $S^1$ (the solid Klein bottle). The exceptional manifolds admit a hyperbolic structure but admit no hyperbolic structures of finite volume.
Here $K^2$ is the Klein bottle.
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
answered Mar 15 at 17:41
Moishe KohanMoishe Kohan
48k344110
48k344110
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3146985%2funderstanding-a-comment-by-thurston%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
I think he just forgot his example. He also forgot few non-orientable examples.
$endgroup$
– Moishe Kohan
Mar 14 at 0:38
$begingroup$
I also think that's the case, specially since this manifold does not show up in the JSJ decomposition of irreducible manifolds with torus boundary, so he really might overlooked it...
$endgroup$
– LLima
Mar 14 at 3:35
$begingroup$
I doubt he forgot it or overlooked it, more likely he had some extra adjectives in his head which didn't make it to his fingers as he typed. Adjectives such as: "orientable"; and "with incompressible boundary".
$endgroup$
– Lee Mosher
Mar 14 at 15:20