Trees and Reflection Groups

We define a reflection in a tree as an involutive automorphism whose set of fixed points is exactly a complete geodesic. We consider the group $\mathrm{Aut}\left( \Gamma \right) $ of automorphism of a tree $\Gamma $ with the topology of uniform convergence in compact sets. With this topological structure, we prove that, for the case of a regular tree of degree $4k$, the group generated by reflections is dense in the group of automorphism with even translation function. It follows that the topological closure of the group generated by reflections has index $2$ in $\mathrm{Aut}\left( \Gamma \right) $.