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 $x\in X$ is compact. How would you construct Euclidean and or spherical geometry using Thurston's definition?

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    $\begingroup$ 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

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