Jon González-Sánchez, Benjamin Klopsch
April 17, 2009
According to Lazard, every p -adic Lie group contains an open pro- p subgroup which is saturable. This can be regarded as the starting point of p -adic Lie theory, as one can naturally associate to every saturable pro- p group G a Lie lattice L ( G ) over the p -adic integers. Essential features of saturable pro- p groups include that they are torsion-free and p -adic analytic. In the present paper we prove a converse result in small dimensions: every torsion-free p -adic analytic pro- p group of dimension less than p is saturable. This leads to useful consequences and interesting questions. For instance, we give an effective classification of 3-dimensional soluble torsion-free p -adic analytic pro- p groups for p > 3. Our approach via Lie theory is comparable with the use of Lazard's correspondence in the classification of finite p -groups of small order.