Finite Contractions of Graphs with Polynomial Growth

Let X be a locally finite, vertex-transitive, infinite graph with polynomial growth. Then there exists a quotient group of Aut(X) which contains a finitely generated nilpotent subgroup N which has the same growth rate as X. We show that X contains a subgraph which is finitely contractible onto theh -dimensional lattice, where h is the Hirsch number of N.