ON the STRUCTURE of GRAPHS WHICH ARE LOCALLY INDISTINGUISHABLE from A LATTICE

For each integer d ≥ 3, we obtain a characterization of all graphs in which the ball of radius 3 around each vertex is isomorphic to the ball of radius 3 in L^d , the graph of the d-dimensional integer lattice. The finite, connected graphs with this property have a highly rigid, 'global' algebraic structure; they can be viewed as quotient lattices of L^d in various compact d-dimensional orbifolds which arise from crystallographic groups. We give examples showing that 'radius 3' cannot be replaced by 'radius 2', and that 'orbifold' cannot be replaced by 'manifold'. In the d = 2 case, our methods yield new proofs of structure theorems of Thomassen ['Tilings of the Torus and Klein bottle and vertex-transitive graphs on a fixed surface', Trans. Amer. Math. Soc. 323 (1991), 605-635] and of Márquez et al. ['Locally grid graphs: classification and Tutte uniqueness', Discrete Math. 266 (2003), 327-352], and also yield short, 'algebraic' restatements of these theorems. Our proofs use a mixture of techniques and results from combinatorics, geometry and group theory.