Non-minimal Tree Actions and the Existence of Non-uniform Tree Lattices

A uniform tree is a tree that covers a finite connected graph. Let X be any locally finite tree. Then G = Aut(X) is a locally compact group. We show that if X is uniform, and if the restriction of G to the unique minimal G-invariant subtree X0 ⊆ X is not discrete then G contains non-uniform lattices; that is, discrete subgroups Γ for which Γ\G is not compact, yet carries a finite G-invariant measure. This proves a conjecture of Bass and Lubotzky for the existence of non-uniform lattices on uniform trees.