Abstract
Let X be a locally finite tree, and
. Then G is a locally compact group. A non-uniform X-lattice is a discrete subgroup
such that the quotient graph of groups
is infinite but has finite covolume, and a non-uniform G-lattice is a discrete subgroup
such that
is not compact yet has a finite G-invariant measure. We show that if X has a unique end and if G contains a non-uniform X-lattice, then G contains a non-uniform G-lattice if and only if any path directed towards the end of the edge-indexed quotient of X has unbounded index.