LATTICES ON PARABOLIC TREES

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.

ACKNOWLEDGMENTS

This first author was supported in part by NSF grant #DMS-9800604. The second author was supported by a VIGRE summer research grant for undergra duates at Harvard University.