Totally disconnected contraction groups

An automorphism alpha of a topological group is called contractive if all alpha-orbits converge to the neutral element. A pair of a locally compact group and a contractive automorphism is called a locally compact contraction group. As shown by Siebert, every locally compact contraction group is a direct product of a simply connected nilpotent Lie group and a totally disconnected, locally compact contraction group.
In the talk, I shall describe results concerning the latter groups, obtained in joint work with George A. Willis. The most recent results concern the structure of contraction groups which are torsion groups.