Automorphism group of a topological space

Question

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

According to the thread http://mathoverflow.net/questions/37356/realizing-groups-as-automorphism-groups-of-graphs on MathOverflow, the answer is yes. — Jeremy Rickard, Nov 25 '14 at 08:36
@JeremyRickard I don't understand; the topological automorphisms of a graph is a big group, much bigger than the group of graph theoretic automorphisms. — , 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. — , Nov 25 '14 at 19:30

score 3 · Accepted Answer · edited Apr 13 '17 at 12:58

As pointed out in the comments, this has been answered by Tony Huynh on MathOverflow. In

de Groot, J. ($1959$), Groups represented by homeomorphism groups, Mathematische Annalen $138$

the author shows that:

"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 $M$.

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