Mixing
Geometry as a Group Action
3
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At 38:45 in this lecture by Thurston he defines a geometry as a an action by a group $G$ on a simply connected topological space $X$ such that the action is transitive and the stabilizer of a point $xin X$ is compact. How would you construct Euclidean and or spherical geometry using Thurston's definition?
group-theory manifolds low-dimensional-topology
asked Jan 28 at 23:56
BobBob
684410
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add a comment |
3
$begingroup$
At 38:45 in this lecture by Thurston he defines a geometry as a an action by a group $G$ on a simply connected topological space $X$ such that the action is transitive and the stabilizer of a point $xin X$ is compact. How would you construct Euclidean and or spherical geometry using Thurston's definition?
group-theory manifolds low-dimensional-topology
asked Jan 28 at 23:56
BobBob
684410
$endgroup$
2
$begingroup$
I think you left out analytic. For Euclidean, $X$ is the plane and $G$ is the group of congruence transformations. For spherical, $X$ is the two-sphere $G$ is $SO(3)$. Read his notes, you can download them from MSRI.
$endgroup$
– Charlie Frohman
Jan 29 at 2:22
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@CharlieFrohman OK thanks. I'll take a look.
$endgroup$
– Bob
Jan 29 at 18:47
$begingroup$
@CharlieFrohman In what branch of mathematics would they introduce the definition of a geometry in this way? I'd like to get a good book on the subject.
$endgroup$
– Bob
Feb 7 at 18:39
$begingroup$
Low dimensional topology. You can download Thurston’s notes from MSRI. The approach goes back to the 19th century. The section with this definition is: library.msri.org/books/gt3m/PDF/3.pdf
$endgroup$
– Charlie Frohman
Feb 7 at 22:05
$begingroup$
Here is an undergraduate introduction: Low-Dimensional Geometry (Student Mathematical Library: IAS/Park City Mathematical Subseries) amazon.com/dp/082184816X/ref=cm_sw_r_cp_api_i_-YkxCbJ7WQR22
$endgroup$
– Charlie Frohman
Feb 7 at 22:11
add a comment |
3
3
3
$begingroup$
At 38:45 in this lecture by Thurston he defines a geometry as a an action by a group $G$ on a simply connected topological space $X$ such that the action is transitive and the stabilizer of a point $xin X$ is compact. How would you construct Euclidean and or spherical geometry using Thurston's definition?
group-theory manifolds low-dimensional-topology
asked Jan 28 at 23:56
BobBob
684410
$endgroup$
At 38:45 in this lecture by Thurston he defines a geometry as a an action by a group $G$ on a simply connected topological space $X$ such that the action is transitive and the stabilizer of a point $xin X$ is compact. How would you construct Euclidean and or spherical geometry using Thurston's definition?
group-theory manifolds low-dimensional-topology
group-theory manifolds low-dimensional-topology
asked Jan 28 at 23:56
BobBob
684410
asked Jan 28 at 23:56
BobBob
684410
asked Jan 28 at 23:56
BobBob
684410
asked Jan 28 at 23:56
BobBob
684410
asked Jan 28 at 23:56
BobBob
684410
684410
2
$begingroup$
I think you left out analytic. For Euclidean, $X$ is the plane and $G$ is the group of congruence transformations. For spherical, $X$ is the two-sphere $G$ is $SO(3)$. Read his notes, you can download them from MSRI.
$endgroup$
– Charlie Frohman
Jan 29 at 2:22
$begingroup$
@CharlieFrohman OK thanks. I'll take a look.
$endgroup$
– Bob
Jan 29 at 18:47
$begingroup$
@CharlieFrohman In what branch of mathematics would they introduce the definition of a geometry in this way? I'd like to get a good book on the subject.
$endgroup$
– Bob
Feb 7 at 18:39
$begingroup$
Low dimensional topology. You can download Thurston’s notes from MSRI. The approach goes back to the 19th century. The section with this definition is: library.msri.org/books/gt3m/PDF/3.pdf
$endgroup$
– Charlie Frohman
Feb 7 at 22:05
$begingroup$
Here is an undergraduate introduction: Low-Dimensional Geometry (Student Mathematical Library: IAS/Park City Mathematical Subseries) amazon.com/dp/082184816X/ref=cm_sw_r_cp_api_i_-YkxCbJ7WQR22
$endgroup$
– Charlie Frohman
Feb 7 at 22:11
add a comment |
2
$begingroup$
I think you left out analytic. For Euclidean, $X$ is the plane and $G$ is the group of congruence transformations. For spherical, $X$ is the two-sphere $G$ is $SO(3)$. Read his notes, you can download them from MSRI.
$endgroup$
– Charlie Frohman
Jan 29 at 2:22
$begingroup$
@CharlieFrohman OK thanks. I'll take a look.
$endgroup$
– Bob
Jan 29 at 18:47
$begingroup$
@CharlieFrohman In what branch of mathematics would they introduce the definition of a geometry in this way? I'd like to get a good book on the subject.
$endgroup$
– Bob
Feb 7 at 18:39
$begingroup$
Low dimensional topology. You can download Thurston’s notes from MSRI. The approach goes back to the 19th century. The section with this definition is: library.msri.org/books/gt3m/PDF/3.pdf
$endgroup$
– Charlie Frohman
Feb 7 at 22:05
$begingroup$
Here is an undergraduate introduction: Low-Dimensional Geometry (Student Mathematical Library: IAS/Park City Mathematical Subseries) amazon.com/dp/082184816X/ref=cm_sw_r_cp_api_i_-YkxCbJ7WQR22
$endgroup$
– Charlie Frohman
Feb 7 at 22:11
2
2
$begingroup$
I think you left out analytic. For Euclidean, $X$ is the plane and $G$ is the group of congruence transformations. For spherical, $X$ is the two-sphere $G$ is $SO(3)$. Read his notes, you can download them from MSRI.
$endgroup$
– Charlie Frohman
Jan 29 at 2:22
$begingroup$
I think you left out analytic. For Euclidean, $X$ is the plane and $G$ is the group of congruence transformations. For spherical, $X$ is the two-sphere $G$ is $SO(3)$. Read his notes, you can download them from MSRI.
$endgroup$
– Charlie Frohman
Jan 29 at 2:22
$begingroup$
@CharlieFrohman OK thanks. I'll take a look.
$endgroup$
– Bob
Jan 29 at 18:47
$begingroup$
@CharlieFrohman OK thanks. I'll take a look.
$endgroup$
– Bob
Jan 29 at 18:47
$begingroup$
@CharlieFrohman In what branch of mathematics would they introduce the definition of a geometry in this way? I'd like to get a good book on the subject.
$endgroup$
– Bob
Feb 7 at 18:39
$begingroup$
@CharlieFrohman In what branch of mathematics would they introduce the definition of a geometry in this way? I'd like to get a good book on the subject.
$endgroup$
– Bob
Feb 7 at 18:39
$begingroup$
Low dimensional topology. You can download Thurston’s notes from MSRI. The approach goes back to the 19th century. The section with this definition is: library.msri.org/books/gt3m/PDF/3.pdf
$endgroup$
– Charlie Frohman
Feb 7 at 22:05
$begingroup$
Low dimensional topology. You can download Thurston’s notes from MSRI. The approach goes back to the 19th century. The section with this definition is: library.msri.org/books/gt3m/PDF/3.pdf
$endgroup$
– Charlie Frohman
Feb 7 at 22:05
$begingroup$
Here is an undergraduate introduction: Low-Dimensional Geometry (Student Mathematical Library: IAS/Park City Mathematical Subseries) amazon.com/dp/082184816X/ref=cm_sw_r_cp_api_i_-YkxCbJ7WQR22
$endgroup$
– Charlie Frohman
Feb 7 at 22:11
$begingroup$
Here is an undergraduate introduction: Low-Dimensional Geometry (Student Mathematical Library: IAS/Park City Mathematical Subseries) amazon.com/dp/082184816X/ref=cm_sw_r_cp_api_i_-YkxCbJ7WQR22
$endgroup$
– Charlie Frohman
Feb 7 at 22:11
add a comment |
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2
$begingroup$
I think you left out analytic. For Euclidean, $X$ is the plane and $G$ is the group of congruence transformations. For spherical, $X$ is the two-sphere $G$ is $SO(3)$. Read his notes, you can download them from MSRI.
$endgroup$
– Charlie Frohman
Jan 29 at 2:22
$begingroup$
@CharlieFrohman OK thanks. I'll take a look.
$endgroup$
– Bob
Jan 29 at 18:47
$begingroup$
@CharlieFrohman In what branch of mathematics would they introduce the definition of a geometry in this way? I'd like to get a good book on the subject.
$endgroup$
– Bob
Feb 7 at 18:39
$begingroup$
Low dimensional topology. You can download Thurston’s notes from MSRI. The approach goes back to the 19th century. The section with this definition is: library.msri.org/books/gt3m/PDF/3.pdf
$endgroup$
– Charlie Frohman
Feb 7 at 22:05
$begingroup$
Here is an undergraduate introduction: Low-Dimensional Geometry (Student Mathematical Library: IAS/Park City Mathematical Subseries) amazon.com/dp/082184816X/ref=cm_sw_r_cp_api_i_-YkxCbJ7WQR22
$endgroup$
– Charlie Frohman
Feb 7 at 22:11