Groups of proper homotopy equivalences of graphs and Nielsen Realization

TL;DR

This paper studies the group of proper homotopy equivalences of graphs, establishing a topology, and proves a Nielsen Realization theorem that connects group actions with graph isomorphisms.

Contribution

It introduces a natural topology on the group of proper homotopy equivalences of graphs and proves a Nielsen Realization theorem for compact subgroups.

Findings

The group of proper homotopy equivalences has a natural Polish group topology.

The Nielsen Realization theorem is established for these groups.

Proper homotopy equivalences can be realized by graph isomorphisms under certain conditions.

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 $Y$.