Groups of proper homotopy equivalences of graphs and Nielsen Realization

For a locally finite connected graph X we consider the group $Maps(X)$ of proper homotopy equivalences of X. We show that it has a natural Polish group topology, and we propose these groups as an analog of big mapping class groups. We prove the Nielsen Realization theorem: if H is a compact subgroup of $Maps(X)$ then X is proper homotopy equivalent to a graph Y so that H is realized by simplicial isomorphisms of Y.