I'm reading the geometric group Theory notes by Bowditch. Assume X is a complete, locally compact geodesic space. A group of isometries G acts on X properly discontinuously. Then X/G is Hausdorff, complete and locally compact. Suppose some orbit is cobounded. How can I show the quotient is compact? A metric space is compact iff it's complete and totally bounded. We already know the quotient is complete. So one approach is to show the quotient is totally bounded. I would appreciate if you can give me a hint on showing the quotient to be totally bounded or provide a hint on another approach.
2026-03-27 06:51:16.1774594276
Geometric Group Theory: cobounded orbit implies cocompact action
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Hint: In a complete geodesic locally compact metric space closed balls are compact, this is a theorem due to Cohn-Vossen, see here. I do not know if your textbook covers this result but the linked Urs Lang's notes provided a self contained proof. Now, what do you know about continuous images of compact topological spaces?