Non-uniform lattices on uniform trees

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 G is not discrete and acts minimally on X, 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 H. Bass and A. Lubotzky ([BL]) for the existence of non-uniform lattices on uniform trees. Our proof is constructive; we produce a non-uniform lattice Γ in G by constructing an infinite graph of finite groups with 'finite volume' which completely determines Γ. We first construct the 'edge-indexed' quotient graph (A, i ) = I(Γ\\X) of X modulo Γ satisfying certain necessary conditions, and we then obtain Γ as a finite 'grouping' of (A, i). This technique for constructing discrete groups which act on trees is naturally suggested by the Bass-Serre theory and was first proposed in [BK]. Our results show, in an explicit way, that the quotient graph Γ\X may have infinitely many cusps of arbitrary geometry, as is indicated by [BL]. The non-uniform tree lattices Γ are known not to have Kazhdan's property T ([VH]), to have arbitrarily large finite subgroups, are not virtually torsion free and cannot be finitely generated ([BL]).