ON EXPONENTIAL GROUPS

Let G be a simply connected solvable analytic group. We say that G is an exponential group if its exponential map is bijective. We say that G is an exponential group with real eigenvalues if Ad g has only real eigenvalues for every g in G, where Ad is the adjoint representation o f G. By a lattice in G, we mean a discrete subgroup GAMMA of G such th at G/GAMMA is compact. We denote the collection of all lattices in G b y L(G) with the Chabauty topology induced from limit of lattices. Let A(G) be the group of all topological automorphisms of G. Equipped with the compact-open topology, A(G) acts on L(G) continuously. We prove t hat for every exponential group G with real eigenvalues and for every lattice GAMMA in G, the orbit A(G)GAMMA is locally compact, and we giv e a counterexample to the case when G is merely an exponential group.