2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/113907We 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.9 pagesGeometric Topology20F65, 20F28text