Locally compact groups with few orbits under automorphisms

The number of orbits under the group of all topological automorphisms was introduced as a new cardinal invariant for topological groups. For locally compact groups, the impact of bounds for this number is investigated. Compact groups with finitely many orbits under automorphisms are pro-finite groups with an open solvable characteristic subgroup; detailed information is obtained for the case of at most 4 orbits. Connected locally compact groups with countably many orbits are simply connected nilpotent Lie groups. Abelian locally compact groups with finitely many orbits under automorphisms are understood quite well. Several open problems are formulated.