The present article continues the study of median groups initiated in [6, 9,
10]. Some classes of median groups are introduced and investigated with a
stress upon the class of the so called A-groups which contains as remarkable
subclasses the lattice ordered groups and the right-angled Artin groups. Some
general results concerning A-groups are applied to a systematic study of the
arboreal structure of right-angled Artin groups. Structure theorems for
foldings, directions, quasidirections and centralizers are proved.