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For a locally finite connected graph $X$ we consider the group $Maps(X)$ ofproper homotopy equivalences of $X$. We show that it has a natural Polish grouptopology, 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 $H$ isrealized by simplicial isomorphisms of $Y$.

Groups of proper homotopy equivalences of graphs and Nielsen Realization