Lattices on Non-uniform Trees

Let X be a locally finite tree, and let G = Aut(X). Then G is a locally compact group. We show that if X has more than one end, and if G contains a discrete subgroup Γ such that the quotient graph of groups Γ\\X is infinite but has finite covolume, then G contains a non-uniform lattice, that is, a discrete subgroup Λ such that Λ\G is not compact, yet has a finite G-invariant measure. 0. Notation, preliminaries and results Let X be a locally finite tree, and G = Aut(X). Then G is naturally a locally compact group with compact open vertex stabilizers Gx, x ∈ V X ([BL], (3.1)). A subgroup Γ ≤ G is discrete if and only if Γx is a finite group for some (hence for every) x ∈ V X. Let μ be a (left) Haar measure on G. By a G-lattice we mean a discrete subgroup Γ ≤ G = Aut(X) such that Γ\G has finite measure μ(Γ\G). We call Γ a uniform G-lattice if Γ\G is compact, and a non-uniform G-lattice if Γ\G is not compact yet has finite invariant measure. Let H ≤ G be a closed subgroup. We may also refer to H-lattices, that is, discrete subgroups Γ ≤ H such that Γ\H has finite measure. A discrete subgroup Γ ≤ G is called an X-lattice if V ol(Γ\\X) := ∑ x∈V (Γ\X) 1 |Γx| is finite, a uniform X-lattice if Γ\X is a finite graph, and a non-uniform lattice if Γ\X is infinite but V ol(Γ\\X) is finite. Bass and Kulkarni have shown ([BK]) that G = Aut(X) contains a uniform X-lattice if and only if X is the universal covering of a finite connected graph, or equivalently, that G is unimodular, and G\X is finite. In this case, we call X a uniform tree. In case G\X is infinite we call X a non-uniform tree. When G is unimodular, μ(Gx) is constant on G-orbits, so we can define ([BL], (1.5)): μ(G\\X) := ∑