The isometry group of a metric space is a topological group (with the compact open topology). The isometry group of a Riemann Manifold is a Liegroup. (Thm. of Steenrod-Myers)


So, is every topological group isomorphic (in the category of topological groups) to the isometry group of a metric space?


And what about the differentiable version: Is every Liegroup isomorphic (in the category of Liegroups) to a isometry group of a Riemann manifold?


Edit: Benjamin Steinberg gave a reference, which fully answers the question in the topological case.
Ryan Budney gave an idea how to realize at least every compact lie group as the isometry group of a Riemannian manifold. (This is proven in [THE ISOMETRY GROUPS OF MANIFOLDS
AND THE AUTOMORPHISM GROUPS OF DOMAINS, by RITA SAERENS and WILLIAM R. ZAME](http://www.ams.org/journals/tran/1987-301-01/S0002-9947-1987-0879582-X/S0002-9947-1987-0879582-X.pdfam=bv.44697112,d.bGE&cad=rja))


So for me, still open is the question about the non-compact case: **Is even every non-compact Lie group realizable as the isometrygroup of a riemannian manifold?**



Is every topological (resp. Lie-) group the isometrygroup of a metric space (resp. Riemannian manifold)?
This seems to be completely answered in the topological category by Thm 1.4 of <http://arxiv.org/pdf/1202.3368v3.pdf>. 


**Edit**


If X is a topological space which is not Dieudonne complete (meaning its topology cannot be given by a complete uniformity), then it seems theorem 7.24 of <http://books.google.com/books?id=v3_PVdvJek4C&pg=PA35&lpg=PA35&dq=free+topological+group+dieudonne+complete&source=bl&ots=QIt0C2TCjN&sig=d4BJjO2h3zM4jDLoTzijrDVYt3w&hl=en&sa=X&ei=VIw8UeuBJpO30QHek4C4AQ&ved=0CC4Q6AEwAA> and Thm 1.4 of the paper above shows that the free topological group on X is not the isometry group of a metric space. Googling shows completely regular spaces exist which are not Dieudonne. 


I believe that any polish group or locally compact group is an isometry group of a metric space by the paper I linked. 


Also the author of the first paper has shown that every Lie group is the isometry group of another Lie group with respect to some proper metric. <http://arxiv.org/pdf/1201.5675v2.pdf>


