# Geometry as a Group Action

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?

• 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. – Charlie Frohman Jan 29 at 2:22
• @CharlieFrohman OK thanks. I'll take a look. – Bob Jan 29 at 18:47
• @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. – Bob Feb 7 at 18:39
• 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 – Charlie Frohman Feb 7 at 22:05
• 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 – Charlie Frohman Feb 7 at 22:11