Groups of proper homotopy equivalences of graphs and Nielsen Realization

Abstract

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 $Y$.