THOMPSON'S GROUP IS DISTORTED IN THE THOMPSON-STEIN GROUPS

We show that the inclusion map of the generalized Thompson groups F(n(i)) is exponentially distorted in the Thompson-Stein groups F(n(1), ... , n(k)) for > 1. One consequence is that F is exponentially distorted in F(n(1), ... ,n(k)) for k > 1 whenever n(i) = 2(m) for some m (whenever no i, m exist such that n(i) = 2(m), there is no obviously "natural" inclusion map of F into F(n(1), ..., n(k))). This is the first known example in which the natural embedding of one of the Thompson-type groups into another is not quasi-isometric.