Groups and Dynamics

Given a connected graph D with v vertices and with a fixed integral second largest eigenvalue r having multiplicity f, one can define an integer lattice L of signature (g, 1) where g = v - f - 1, such that the symmetries of D act on L. This kind of construction goes back to atleast [Neumaier and Seidel, 1983] and is analogous to building a root lattice from a Dynkin diagram. If D is strongly regular, then this construction contains the Euclidean representation of D which is an useful tool in studying existence questions of strongly regular graphs with specified parameters. The aim of this talk is to illustrate that this construction and a hermitian variant of it produces interesting Loterntzian lattices and interesting complex reflection groups in U(g, 1). In particular, an example in U(4, 1) is related to the moduli space of cubic surfaces and an example in U(13, 1) is conjecturally related to the monster simple group.