I am studying Svarc-Milnor lemma :
If a group $G$ acts geometrically on a proper geodesic metric space $(X,d)$, then $G$ is finitely generated and any of its Cayley graphs (with some finite generating subset) is quasi-isometric to $(X,d)$.
I would like to check with examples the necessity of each hypothesis in the lemma. In particular, I'm having trouble finding a geometric action of a non-finitely generated group on a non-proper geodesic metric space.
I was thinking about the space $l_2$ (of sequences) which is not proper, but I don't find any geometric action with it (for example with reflexions, it is not cocompact).
Note : geometrically means that the action is by isometry, properly discontinuous and cocompact (the set of orbits is compact). The properly discontinuity means here that for every compact $K \subset X$, there exists only a finite number of $g \in G$ such that $ g \cdot K$ intersects $K$.
Thank you.
Best Answer
Yes, one can drop the properness (of $X$) assumption in the M-S lemma. You can find a proof (a bit sketchy but correct) in
Roe, John, Lectures on coarse geometry, University Lecture Series 31. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3332-4/pbk). vii, 175 p. (2003). ZBL1042.53027.
Specifically, see pages 8-10.