Exponential radicals of solvable Lie groups

For any connected Lie group G, we introduce the notion of exponential radical Exp(G) that is the set of all strictly exponentially distorted elements of G. In case G is a connected simply-connected solvable Lie group, we prove that Exp(G) is a connected normal Lie subgroup in G and the exponential radical of the quotient group G/Exp(G) is trivial. Using this result, we show that the relative growth function of any subgroup in a polycyclic group is either polynomial or exponential. (C) 2002 Elsevier Science (USA).