On exponential groups

AbstractLet 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 of G. By a lattice in G, we mean a discrete subgroup Γ of G such that GΓ is compact. Wedenote the collection of all lattices in G by 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 that for every exponential group G with real eigenvalues and for every lattice Γ in G, the orbit A(G)Γ is locally compact, and we give a counterexample to the case when G is merely an exponential group