We consider low-dimensional groups and group-actions that are definable in a supersimple theory of finite rank. We show that any rank 1 unimodular group is -by-finite, and that any 2-dimensional asymptotic group is soluble-by-finite. We obtain a field-interpretation theorem for certain measurable groups, and give an analysis of minimal normal subgroups and socles in groups definable in a supersimple theory of finite rank where infinity is definable. We prove a primitivity theorem for measurable group actions.