A note on the exponential map of a real or \(p\)-adic Lie group (Q1339392)

A real (resp. \(p\)-adic) Lie group \(G\) is said to have the additive exponential property AEP if for all (resp. all small) elements \(X,Y\) of the Lie algebra \({\mathfrak g}\) of \(G\) the equation \(\exp X \cdot \exp Y = \exp (X + Y)\) implies \([X,Y] = 0\). The group \(G\) is said to have the multiplicative exponential property MEP if \(\exp X \cdot \exp Y = \exp Y \cdot \exp X\) implies \([X,Y] = 0\). The authors show that a connected real Lie group has AEP exactly if it is nilpotent and has the property that \([X,Y] \in {\mathfrak z} ({\mathfrak g})\) and \(\exp ({1 \over 2} [X,Y]) = 1\) imply \([X,Y] = 0\). Consequently, every simply connected (or, more generally, every linear) connected nilpotent real Lie group has AEP. In the \(p\)-adic case, the picture is quite different. For instance, it is proved that the group of \(\mathbb{Q}_p\)-rational points of any algebraic group defined over \(\mathbb{Q}_p\) has AEP on \({\mathfrak g}_0\), if only \(p > 3\). Here \({\mathfrak g}_0\) denotes the Lie ring of all ``small'' elements of \({\mathfrak g}\) (that is, those where a given valuation takes values greater than \((p - 1)^{-1})\). A connected nilpotent real Lie group has MEP exactly if it has AEP. However, there are nonnilpotent groups with MEP. Every connected real Lie group with MEP is solvable, and its adjoint representation has no purely imaginary roots. In the simply connected case, the latter condition implies the first. Finally, the authors prove that all \(p\)-adic linear groups have MEP.