  We show that the group of all pl-homeomorphisms of the reals having bounded
slopes surjects on the group $QI({\Bbb R})$ of all quasi-isometries of ${\Bbb
R}$. We prove that the following groups can be imbedded in $QI({\Bbb R})$: The
group of compactly supported pl-homeomorphisms of the reals, the Richard
Thompson group F, and the free group of rank the continuum.
