# Automorphism group of a topological space

Let $G$ be any group. Is there a topological space $(X,\tau)$ such that the automorphism group $\textrm{Aut}(X,\tau)$ is isomorphic to $G$?

• At least for finite groups it is true. – Dune Nov 25 '14 at 8:13
• According to the thread mathoverflow.net/questions/37356/… on MathOverflow, the answer is yes. – Jeremy Rickard Nov 25 '14 at 8:36
• @JeremyRickard I don't understand; the topological automorphisms of a graph is a big group, much bigger than the group of graph theoretic automorphisms. – user98602 Nov 25 '14 at 18:02
• @MikeMiller The thread I pointed to contains a reference to a construction for topological spaces as well as one for graphs: de Groot, J. (1959), Groups represented by homeomorphism groups, Mathematische Annalen 138 – Jeremy Rickard Nov 25 '14 at 19:29
• @JeremyRickard Ah, thanks. Sorry, don't know how I missed that. – user98602 Nov 25 '14 at 19:30

de Groot, J. ($1959$), Groups represented by homeomorphism groups, Mathematische Annalen $138$
"for every group $G$ one can find a complete, connected, locally connected metric space $M$ of any positive dimension such that $G \cong A(M)$"
where $A(M)$ denotes the autohomeomorphism group of $M$.